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CQEmiGm DEPOSIT. 



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SHEET METAL DRAFTING 






''MlBockQxIr, 



'rCM'J-llii DOOi LU Jm 

PUBLISHERS OF BOOKS F O FL^ 

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INDUSTRIAL EDUCATION SERIES 



SHEET METAL DRAFTING 



PREPAEED IN THE 

EXTENSION DIVISION OF 
THE UNIVERSITY OF WISCONSIN 



'BY 
ELLSWORTH M. LONGFIELD 

HEAD, SHEET METAL DEPARTMENT, 
BOSTON TRADE SCHOOL 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 

NEW YORK: 370 SEVENTH AVENUE 
LONDON: 6 & 8 BOUVERIE ST., E. C. 4 

1921 









Copyright 1921, by the 
McGkaw-Hill Book Company, Inc. 



0)CU6il7.58 



I\PR 21 1321 



^ t C f 



tZi^' 



PREFACE 



This text on "Sheet Metal Drafting" was prepared especially 
for correspondence-study instruction in the Extension Division of 
the University of Wisconsin. It is also admirably adapted as a 
text book for Vocational, Evening, and Part-time Schools. 

The underlying principles of sheet metal pattern drafting are 
presented, each chapter discussing a different principle. The 
sequence of principles has been arranged with due regard to the 
well-known factors governing the student's progress through such 
a course of instruction. This arrangement has been successfully 
tested by several years of practical application in the teaching of 
the subject. 

The problems in "Related Mathematics" point out the appHca- 
tions of mathematical principles to sheet metal work and serve as a 
guide for the proper correlation of the work in mathematics, 
drawing, and shop practice. 

When the text is used for vocational, evening, and part-time 
schools, the articles considered in the various chapters can be 
manufactured in the shop school. When these are sold, the cost 
of instruction is considerably reduced without sacrificing the edu- 
cational value of the course. 

Acknowledgments are due Ben G. Elhott, M. E., Professor of 
Mechanical Engineering, University of Wisconsin, for his sug- 
gestions as to the general form and content of the text, and valu- 
able editorial work. 

Ellsworth M. Longfield. 

February, 1921. 



CONTENTS 



CHAPTER I 

Rectilinear Figures 

ARTICLE PAGE 

1. Sheet Metal Drafting 1 

2. Orthographic Projection 2 

3. Drawing Instruments 3 

4. Lines 7 

5. Dimensions 8 

6. Lettering 9 

7. Titles 9 

8. Filing Circles .10 

9. The Sheet Metal Cleat 12 

10. Related Mathematics on Sheet Metal Cleat 13 

11. The Galvanized Match Box 14 

12. Related Mathematics on Galvanized Match Box 16 

13. The Candy Pan 17 

14. Related Mathematics on Candy Pan 19 

15. The Iron Bread Pan 21 

16. Related Mathematics on Iron Bread Pan 23 



CHAPTER II 
Wired Cylinders 

17. The Chironey Tube 30 

18. Related Mathematics on Chimney Tube 32 

19. Half-pint Cup 34 

20. Related Mathematics on Half-pint Cup 36 

21. The Painter's PaU 38 

22. Related Mathematics on Painter's Pail 40 

23. The Garbage Can 41 

24. Related Mathematics on Garbage Can 42 

vii 



viii CONTENTS 



CHAPTER III 

Cylinders Cut by Planes 
article page 

25. The Scoop 46 

26. The Two-piece Elbow 49 

27. The Two-piece 60° Elbow 52 

28. The Four-piece 90° Elbow 55 

29. The Long Radius Elbows 57 

30. Related Mathematics on Elbows ,,58 

31. The Backset Method 63 



CHAPTER IV 

Intersecting Cylinders 

32. Tee Joint at Right Angles 66 

33. The Tangent Tee at Right Angles . 69 

34. The Tee Joint Not at Right Angles 72 

35. The Tangent Tee Not at Right Angles 75 

36. Related Mathematics on Tangent Tees and Tee Joints ..... 76 

CHAPTER V 

Cones of Revolution 

37. The Conical Flower Holder 80 

38. Related Mathematics on Conical Flower Holder 82 

39. The Pitch Top Cover 83 

40. Related Mathematics on Pitch Top Cover 85 

41. The Vegetable Rarer 86 

42. The Conical Roof Flange 89 

43. Related Mathematics on Conical Roof Flange . . . . . . .91 

44. The Apron for a Conical Roof Flange 93 

CHAPTER VI 

Intersecting Rectangular Prisms 

45. Three-piece Rectangular Elbow 96 

46. Related Mathematics on Elbows 97 

47. The Rectangular Pipe Offset 100 

48. Related Mathematics on Rectangular Pipe Offset 102 

49. The Diagonal Offset 103 

50. Related Mathematics on Diagonal Offsets ...%.... 105 

51. The Curved Elbow in Rectangular Pipe 106 

52. Related Mathematics on Curved Elbows 106 



CONTENTS ix 



CHAPTER VII 

Planning for Quantity Production 
article page 

53. Planning for Quantity Production of an Ash Barrel 110 

54. Related Mathematics on Ash Barrel 117 

CHAPTER VIII 

Sections Formed by Cutting Planes 

55. The Sprinkling Can 120 

56. Related Mathematics on Sprinkling Can 123 

57. Boat Pump 125 

58. Related Mathematics on Boat Pump 127 

59. The Roof Flange 129 

CHAPTER IX 

Frustums of Rectangular Pyramids 

60. Dripping or Roasting Pan 136 

61. Related Mathematics on Dripping Pans 138 

62. Rectangular Flaring Pans 140 

63. Related Mathematics on Rectangular Flaring Pans . . . . . 142 

64. Register Boxes 143 

65. Related Mathematics on Register Boxes 145 

CHAPTER X 

Combinations of Various Solids 

66. Atomizing Sprayer 148 

67. Related Mathematics on Atomizing Sprayer 150 

68. Ash Pan with Semicircular Back 153 

69. Related Mathematics on Ash Pan 155 

70. Rotary Ash Sifter 156 

71. Related Mathematics on Rotary Ash Sifter 160 

CHAPTER XI 

Frustums op Cones 

72. Cup Strainer 162 

73. Related Mathematics on Cup Strainer 164 

74. Short Handled Dipper 165 

75. Related Mathematics on Short Handled Dipper 167 

76. Liquid Measures 169 

77. Related Mathematics on Liquid Measures 171 



2 SHEET METAL DRAFTING 

2. Orthographic Projection. — Before attempting to make any 
drawings, one must first get a clear idea of the way in which 
objects are represented in mechanical or orthographic drawings. 
If a person is going to make a photograph of an object, he nearly 
always makes a view taken from one corner so as to show as many 
sides as possible in order to give a complete idea of the object in 
one picture. For example, Fig. 1 shows how an anvil would be 
represented in a single view or picture so as to give a complete 
idea of its shape. Such a drawing of most objects would be very 



c 



J. 



h 




Left Eno 
View 



tz 
Side V/ew 




Right End 
View 



Bottom View. 



Fig. 2. — Mechanical Drawing of Anvil. 

complicated or difficult to make, and even then in many cases it 
would not give the complete idea. Instead of making a pictorial 
drawing, the draftsman makes two or more views as if he were 
looking straight at the different sides of the object as in Fig. 2. 
At A is shown what would be called a "front elevation," meaning 
a view of one side taken from the front with the anvil set up in 
its natural position. At D is shown the "plan" or top view. 
This shows what would be seen by looking down on the anvil from 
above along the direction of the arrow Y. At B and C are shown 
the views of the ends as seen by looking along the arrows, W 
and A^. These views are called the "right end elevation" and 



RECTILINEAR FIGURES 3 

"left end elevation," depending upon whether the view is that of 
the right end or the left end. At E is shown the bottom view, 
which would be that obtained by looking up from beneath in the 
direction of the arrow, Z. These views are not all needed to show 
the complete shape of the anvil. They are, however, all the 
different views that might be used by the draftsman. These 
"views" are also called the "projections" of the object, and this 
method of showing it is called "projection." 

Drawings are made in the drafting room and are then sent to 
the shop so that the object shown can be made. Consequently, all 
drawings must have complete information on them so that no 
questions need be asked. Besides showing the shape and size of 




,HEAC> 

UPPER eOGE 



LOWER EDQE 

Fig. 4. — T-Square. Fig. 5. — Testing Straightness 

of Working Edges. 

the parts, the drawings must have full information as to the 
material to be used, its gage, number wanted, etc. 

3. Drawing Instruments. — The Drawing Board and T- 
Square. — The drawing board. Fig. 3, is for the purpose of holding 
the paper while the drawing is being made. It is usually made of 
some soft wood, free from knots and cracks and provided with 
cleats across the back or ends. 

The T-square, Fig. 4, consists of a head and blade fastened 
together at right angles. The upper or working edge of the blade 
is used for drawing all horizontal lines (lines running the long way of 
the board) and must be straight. 

The left end of the drawing board must also be straight. The 
working edge of the T-square and the working edge of the drawing 



SHEET METAL DRAFTING 



board may be tested for straightness by holding them as shown 
in Fig. 5. They should be in contact along their entire length. 
If they are not, one or the other is not straight. 

The working position of the drawing board and T-square is 
illustrated in Fig. 6. In this position the blade of the T-square 
can be moved up and down over the surface of the board with the 
left hand while holding the head firmly against the working edge 




Fig. 6. — Working Position of Drawing Board and T-Square. 

of the board. For a left-handed man, the working edge of the 
board will be at the right, with the head of the T-square held 
firmly in place with the right hand. 

All horizontal lines should he drawn with the T-square, draw- 
ing from left to right, meanwhile holding the head of the T-square 
firmly against the working edge of the hoard with the left hand. 

The Triangles. — The triangles are used for drawing lines other 
than horizontal. They are made of hard rubber, celluloid, wood, 





4-5° triangle 

Fig. 7. 



30 x 60° triangle 
Fig. 8. 



or steel. There are two common shapes, called the 45° triangle 
and the 30°X60° triangle. These are illustrated in Figs. 7 and 8. 
The 45° triangle, shown in Fig 7, has one angle (the one marked 
90°), a right angle. There are 90° in a right angle. The other 
angles are 45° each (just half of a right angle). One angle of the 
30°X60° triangle is a right angle; another angle is 60° (just f of a 
right angle) ; and the other is 30° (just ^ of a right angle). 

For drawing vertical hues (lines at right angles to the horizon- 
tal lines which have already been explained), the T-square should 



RECTILINEAR FIGURES 



be placed in its working position and one of the triangles placed 
against its working edge. In Fig. 9 is shown the correct position 




Fig. 9. — Drawing Vertical Lines by the Use of T-Square and Triangle. 




Fig. 10. — Method of Drawing Various Angles by Use of Triangles. 



of the hands and the method of holding the pencil for drawing 
these lines. 



6 SHEET METAL DRAFTING 

Figure 10 illustrates the method of getting various angles by- 
means of the triangles separately and in combination. These 
angles, of course, can be drawn in the opposite slant by reversing 
the triangles. 

Always keep the working edge of the triangle toward the head 
of the T-square and draw from the bottom up, or away from the body. 

The Pencil. — The pencil must be properly sharpened and kept 
sharp. Good, clean-cut lines cannot be made with a dull pencil. 




' Fig. 11. — Pencil Sharpened in the Proper Way. 

A pencil sharpened in the proper way is shown in Fig. 11. The end 
"a" shows the chisel point which is used for drawing lines; the end 
"6" shows the round point used for marking off distances and for 
putting in dimensions, lettering, etc. About f in. of lead should 
be exposed in making the end "a." Then it should be sharpened 
flat on two sides by rubbing it on a file or piece of sandpaper. 

The Scale. — A scale is used in making a drawing on an ordinary- 
sized sheet of paper, so that the drawing is of the same size as the 
object, or some number of times larger or smaller than the object. 



W\^^^K^V^^W\^^\^\^A^W\^^\;VV^^W\^ \ ^VVW\^V\^V\^\'.W 




Fig. 12. — Triangular Scale. 

The triangular scale illustrated in Fig. 12 has six different scales, 
two on each side. 

The ordinary architect's triangular scale of Fig. 12 has eleven 
scales. On the scale of three inches equals one foot, a space that 
is three inches long is divided into twelve equal parts, each of 
which represents an inch on the reduced scale and is itself sub- 
divided into two, four, eight, or sixteen equal parts, corresponding 
to halves, quarters, eighths, or sixteenths of an inch. The other 
scales are constructed in the same way. A little study of the 
scale with the above description of its construction will make its 
use clear. 



RECTILINEAR FIGURES 7 

4. Lines. — In Fig. 13 the names and uses of various kinds of 
lines which are used in making a drawing are shown. This 
figure also shows a table of the relative weights of these lines. 
Border Lines. — The border line needs no detailed explanation. 
Object or Projection Lines. — The visible object line is a line that 
represents any definite edge that may be seen from the position 
that the observer assumes in obtaining the given view. The invis- 
ible object or projection line represents a line or edge of an object 
that cannot be seen from the observer's point of view but which 
actually exists and may be seen from some other position or point 
of view. For instance, the drawing board cleats on the bottom 



Character of Line 


Name of Line 


Weight or 
Width of Line 




BORDER 

OBJECTOR PR0JECTI0N(V1$1BLE) 

79 » „ (invisible) 
behdingLime 

CENTtR LINE 
EXTENSION LINE 
DIMENSION LINE 
SECTION LINE 
BROKEN MATERIAL 


32 
1" 

64 

1" 

64 
i" 

64 
1" 

iza 
I" 

128 

JLl' 
128 

Jl' 
128 
1" 
64 












\f -.1 


K >1 


/.^^^.^.^^-^w- 



Fig. 13. — Lines Used in Drawing. 



could not be seen from above, but could be seen from the sides or 
when the board was held above the eye. For the sake of deter- 
mining the relation of such cleats to some other member that 
might be required on the top of the board, the cleats would be 
shown by invisible or broken lines on the top view. 

Bending Lines. — The lines that are drawn on a layout or pattern 
of an object to indicate the location of an edge in the completed 
object are called bending lines, since they locate on the layout or 
pattern the line along which a bend must be made. These lines 
differ from the regular object or projection lines only by having at 
each end a small free-hand circle drawn upon them. Bending 
lines are drawn differently by different authors and in some shops, 



8 



SHEET METAL DRAFTING 



but as long as a definite logical system is followed it does not make 
much difference what system it is. 

Center Lines. — Center lines are, in general, lines of symmetry; 
that is, they usually divide the views of an object into two equal 
though not exactly similar parts, since one is right-handed and 
the other left-handed. In some cases, however, the two parts are 
sometimes unequal, as well as dissimilar. 

Center lines are used to aid in dimensioning; to line up two or 
more related views; and to fix definitely the centers of circles. 
All circles have two center lines at right angles to one another, 
usually a vertical and a horizontal center line. 

Extension Lines. — Extension- lines are used to extend object or 
projection lines in order to line up related views and to insert 
dimensions without placing them on the object itself, causing 




IVENTILATING PIPE PATTERN 

"-K^ 10 REQUIRED SHEET IRON 

*-l^^ QUARTER SIZE 

'^i^'VoUR NAME DATE 



BORDER LiriES 



Fig. 14. — Simple Title for Drawing. 



a confusion of lines. They should fail to touch the object by 
about Y6 ill- 

Dimension Lines. — Dimension lines are used with arrowheads 
at each end to show the limits of a given dimension. The lines 
are broken at some point, usually near the center, in order to insert 
the figures. 

5. Dimensions. — Dimensions up to 24 inches are, in general, 
given in inches, as 16|". Above 24 inches, practice varies, but, 
in general, feet and inches are used as 3 -2|", for 3 feet 2| inches. 

Vertical figures about | in. high are usually used for dimensions. 
Fractions should be as large as whole numbers and care should 
be taken to see that the figures do not touch the dividing fine. 
The dividing line of a fraction should be on a level with the dimen- 
sion line as in Fig. 14. 



RECTILINEAR FIGURES 9 

Horizontal dimensions should he read from the bottom of the 
sheet, and vertical dimensions from the right-hand side of the sheet 
as in Fig. 16. 

PREE.HAND LETTERING 



ABCDEFGHIJKLMN0PQR5TUVWXYZ& 
:^/^ 'Bv Xi iS V" 12 3 4 5 6 7S> SO 

ABCDErGHIJKLMNOPQRSTUVWXYZ& 
I234567690 4-| 3| 7^ 

Fig. 15. — Satisfactory Style of Lettering for Sheet Metal Drawings. 

6. Lettering. — Lettering is very important for the draftsman, 
and ability to make good letters is a good asset for anyone. Fig- 
ure 15 shows the type of lettering that is very frequently used and 
that, is probably most easily made. The strokes for forming the 
letters are shown by the arrows. 



-•^ rfi: 



=t^ 



^^ 



-1^ 



PA'i I E- Ki a: 



FOK 



VFNTlLAllNCn PIPF " 



FNfilNKFKING DFPTT 



UNIV. F y r. mv. 



1 HF Univ. OF \a/i«^ 



MAnig,nR^ 



DATE- 9-3-10. CH.-H.C.L 



5CALE-4'=r 



DR-J.E.B. 



TR.-A.C.M. 



APR-H.M.R. 



NO.WANTED-(0. 



mat'l 



#22 GAL. IRON 



/" 



Fig. 16.— Detailed Title Corner. 

7. Titles. — No drawing is complete without a title which 
gives such information as the drawing itself fails to impart. Fig- 
ures 14 and 16 illustrate two different titles, of which the former is 
the simpler. Figure 16 represents a title such as would appear on 
an up-to-date shop drawing made in a large office where it would 
have to be traced, checked, and approved before being blue printed. 



10 SHEET METAL DRAFTING 

8. Filing Circles. — In all well-regulated manufacturing estab- 
lishments some system of filing away the drawings is used so that 
they may be easily found when needed. 

In one such system, filing circles are placed, one at the lower 
left-hand corner of the sheet and another, upside down, at the upper 
right-hand corner, so that no matter how the drawing is placed 
in the drawer, a filing circle will always appear at the lower left- 
hand corner. 

Figure 14 shows the size of the circle to be used, its location on 
the sheet, and three different numbers. The first number on the 
upper line represents the number of the general order; the second 
number is the humber of the detail sheet under this general 
order; and the bottom number is the number of the section or 
drawer in the filing case in which this sheet is to be found. A card 
index is used in connection with this filing system to facilitate the 
location and the handling of the drawings. 



RECTILINEAR FIGURES 



1] 



No 



Job 



Drawing 
Objectives 



Mathematical' 
Objectives 




Cleat 




Match Box 



/dieo of stretch- 
ovf; />rof//e a/ic/ 
front e/etraf/on. 
O/mens/on/n^ file 

draining. 
Accurate meas- 

(/re /vents. 



Def/nif/'o/} of 
'^Oirff]" arref 
"Cut" 

Ad£^i'f/orj of 
fracf/otjs. 



Front &5/df> 
e/e^'ot/'o/JS 
loco ting fyo/es 
tot>e dr/'lled. 
L<jf>ped and 
so/dered y'o/n ts 



Aetd/t/dn of 
fract/'ons. 

fifofh/)/fcaf/or? 

of fracf/on% 
Area of 
recfan<^le. 




Al/oyva/ices 
for itviring 

Conception of 
rectang le 
f/or-Jn^ n/orh 



Pe.c'tmal'i. 

Cf/ttinq stocti 
fo adyan-tat^e,. 

Perce-ntoge of 




Bread 
Pan 



Double Seamed 
Qnds>. 

Notching for 

seawi <S: ir^/re 



Area of 
tra/?exoid 

iVe/<^titofpan. 



Objectives of Problems on Rectilinear Figures. 



12 SHEET METAL DRAFTING 

Problem 1 
LAYING OUT A METAL CLEAT 

9. The Sheet Metal Cleat. — The work of this problem will 
consist in laying out, to full size, the views and pattern for a gal- 
vanized sheet metal cleat. In making the layout for this cleat, 
the following points must be kept in mind: 

1. The proper relation of views in a drawing. 

2. How to dimension a drawing. 

3. Accuracy in the use of the scale rule. 

This cleat is to be formed from a flat piece of No. 16 galvanized 
iron; all the bends to be made to an angle of 90°. Figure 17 
represents the cleat as it would appear on a photograph. 

Before starting the layout, it must first be determined how 
many faces the cleat has. By holding the cleat with the largest 
surface directly in front of the eyes, three of these faces can be 
seen. A drawing should be made of what is actually seen. This 
drawing would appear as shown in Fig. 18. This view is called 
the front elevation and from it the exact sizes of the three faces or 
surfaces shown can be determined. 

If the cleat is turned so that the eyes see the thin edge of the 
metal, the view will be as shown in Fig. 19. This view is called 
the profile because it is the exact shape or outline to which the 
cleat must be formed in the shop. 

In drawing the profile, it can be located directly under the 
front elevation by using extension lines such as shown. In addi- 
tion to showing the exact outline of the cleat, the profile also shows 
the dimensions necessary for laying out the pattern. In order to 
transfer these dimensions to the line of stretchout on. the pattern, 
the profile should be numbered as indicated. 

The front elevation and profile furnish all the information re- 
quired to lay out the pattern. Consequently, there is no need to 
draw other views. 

The line of stretchout is always drawn at right angles to the 
side of the view from which the pattern is to be taken. Upon this 
line of stretchout, all of the distances (called the spacing of the 
profile) of the profile should be placed and numbered to correspond. 
Tliis has been done in Fig. 20. Perpendicular lines are then 
drawn through these numbered points. These are called the 



RECTILINEAR FIGURES 



13 






ELE.VATION 

Fig. 18 




ir 




( 


1 ( ) ( 1 ( 
Line of stretchout 


12 3 4 5 6 
PATTERN 

< I < ' oil 


3 








^ i'"» 


l^ir J 


-'" > 


ll" 


hii"-* 


^ 16 1 16 


' t-8 'is 


' 




"* -l 


PROFILE. 

^ F,al9 ^ 




1 


Fig. 20 


1, '=" 


L, 13".^ 


5 


r-ien 


•-e * 


r"i6 





Figs. 17-20.— Sheet Metal Cleat. 



measuring lines of the stretchout. Extension Hnes carried over 
from the elevation locate the top and bottom lines of the pattern. 
The side lines of the pattern are formed by measuring lines No. 1 
and No. 6. Small free-hand circles should be placed as shown to 
indicate to the workman where the bends are to be made. The 
views and the pattern must be fully dimensioned. 

10. Related Mathematics on Sheet Metal Cleat. — If the 
drawing is correct, the sum of all the lines in the profile will be 
equal to the length of the line of stretchout. 

Problem lA. — Compute the sum of all the lines in the profile 
from the dimensions given in Fig. 17. Measure the length of the 
line of stretchout in Fig. 20 and compare with the sum of the pro- 
file lines. If the answers do not agree, either the drawing or the 
arithmetic is incorrect. They should be made to agree. 



14 SHEET METAL DRAFTING 

Problem 2 
GALVANIZED MATCH BOX 

11. The Galvanized Match Box. — In laying out the views and 
the pattern for the galvanized match box, special attention should 
be directed to the following: 

1. Locating holes accurately. 

2. Showing hidden hems and surfaces. 

3. Showing lapped and soldered joints. 

Figure 21 shows a rectangular match box having the back 
raised above the other upright surfaces. There are two screw 
holes drilled or punched in the back. 

The top edges of the back, the ends, and the front side are 
provided with a -j^-inch hem so that the edge will be smooth. 
The hems on the top of the back and on the right end 
can be seen in Fig. 21, but the hems on the top of the front and 
the left end cannot be seen from the outside because they are 
hidden from view. These hems are shown by dotted lines. The 
dotted line is always used to show the position of a line which 
cannot be seen. All dotted lines in this view represent lines or 
edges which cannot be seen. 

This box has five surfaces, a back, a front, two ends, and a 
bottom. A full size elevation of the end surface will appear as 
in Fig. 22. The hem on the top edge of the end surface is shown 
by a dotted line and the lap by a solid line. This end elevation 
is also a profile view, and dimensions taken from it can be used in 
laying out the Hne of stretchout for the pattern. The end eleva- 
tion is, therefore, numbered in a way similar to that in Fig. 19 of 
Problem 1. 

The front elevation is constructed by using the extension lines 
from the end elevation. This front elevation shows the length 
of the box, together with the location of the two holes in the back. 

As in Fig. 20 of Problem 1, the line of stretchout is drawn at 
right angles to the view from which the pattern is to be taken. 
The line of stretchout should be numbered to correspond to the 
profile numbering in the end elevation. Extension fines dropped 
from the front elevation determine the outside edges of the box on 
the pattern. The 3^-inch hems must be added at the top and the 
bottom of the stretchout fine. The |-inch laps must also be added 



RECTILINEAR FIGURES 



15 




END ELEVATION 
J Lop seom 



151 



I- FH/^'"'" 



front elevation 
Fig. Z3 



-2r6 



Pattern of end 
(two wonted) 



-2|6"— 

FiQ. as 



> 



Pattern of 
(.One w< 

FlQ. 



body 
n+ed) 

2.4 



(w. 



< 



Figs. 21-25.— Galvanized Match Box. 



16 SHEET METAL DRAFTING 

to the remaining edges of the pattern. Free-hand circles must be 
placed to indicate where the bends are to be made. Suitable 
notches are provided at the laps so that they will fit together at 
an angle of 90° at the bottom corners of the box. 

A separate pattern must be drawn for the two ends of the 
box, as these are not included in the main pattern. Extension 
lines dropped from the end elevation will determine the length of 
the end pieces. The height of the end pieces can be taken either 
from the pictorial view in Fig. 21, or from the end elevation. A 
j^-inch hem must be added to the top of the end pattern as shown. 
The ends of the hem are notched slightly as indicated. 

The over-all dimensions of the patterns should be put in as 
indicated by the question marks on the drawing. The end and 
front elevations are to be dimensioned as indicated in Figs. 22 
and 23. 

12. Related Mathematics on Galvanized Match Box. — 
Girth and Cut. — The ''Girth" is the distance around the profile 
view. The "Cut" is the distance around the profile plus the laps 
or locks necessary to join the pieces of metal together. 

Over-all Dimensions. — Dimensions showing the sizes of the 
blank pieces of metal required to "get out " the job should be placed 
upon every pattern. These are known as "over-all" dimensions, 
as they include both the pattern and the edges allowed. Dimen- 
sion lines for this purpose are, indicated on Fig. 24 by question 
marks. 

Rectangle. — A rectangle is a flat surface bounded by four 
straight lines forming right angles at their points of meeting. 
Figures 20 and 24 are examples of rectangles. The area of a rec- 
tangle is equal to the length multiplied hy the width. 

Problem 2 A. — Compute the over-all dimensions from Fig. 22. 
Check these answers by measuring the drawing, and place the 
correct figures on the over-all dimension lines of Figs. 24 and 25. 

Problem 2B. — Find the area of Fig. 24 and also the area of Fig. 
25. (Use over-all dimensions.) 

Problem 2C. — Find the total area of the metal required to con- 
struct the box. (One body and two end pieces.) 



RECTILINEAR FIGURES 17 

Problem 3 
CANDY PAN 

13. The Candy Pan. — As already pointed out in the previous 
problem, the elevation view of an object may sometimes be used as 
a profile for laying out the pattern. This is the case with the 
layout of the tin candy pan considered in this problem. Particu- 
lar attention should be given to the methods of wiring the top of 
this pan. 

The pan shown in Fig. 26 is known to the sheet metal trade 
as a rectangular flaring pan. Flaring is another word for taper- 
ing. Since the sides flare or taper, the bottom of the pan must be 
smaller than the top. This pan has an equal flare on all sides. 
Some pans have an unequal flare; that is, some of the sides 
taper more than others. This candy pan is to be made of 
sheet tin, wired with No. 12 wire, the corners to be lapped and 
soldered. 

A full size elevation, as in Fig. 27, showing one corner broken 
away to reveal the wire, is to be drawn. Care should be taken 
that the flare is equal on both sides. As stated before, this eleva- 
tion also serves as a profile and is numbered 1, 2, 3, and 4. It 
should be noticed, in numbering the profile, that the metal neces- 
sary to cover the wire is not included. This is on account of the 
fact that there is a standard allowance for covering wire. For 
covering a wire with metal, add an edge to the top of the pattern equal 
to 2\, times the diameter of the wire. This allowance, however, must 
be increased for metal heavier than No. 23 gage. 

The plan view is drawn below the elevation as shown in Fig. 
28. The fine of stretchout is laid out at right angles to the long 
side of the plan. The spacing of the profile is laid off on the 
stretchout and is numbered to correspond. The allowance for 
covering the wire must be computed according to the rule given 
above. Number 12 wire has a diameter of approximately ^ in. 
This allowance for wiring is set off to the right of No. 4 and to the 
left of No. 1 on the line of stretchout. The measuring lines of the 
stretchout are then drawn. The extension lines from the plan 
are carried over to the stretchout view and the pattern of one of 
the flaring sides constructed as shown in Fig. 29. Since the flare 
is equal on all sides of the pan the other three sides can be laid out 



18 



SHEET METAL DRAFTING 




FiQ. Z6 



\\*- ^^ \ \ 

-¥"1 ^ Elevotfon /^ ^""-^P 



Fi<3. er 



r^ 



7' 



Plon 
F.Q. E8 



WHO 

T 



seom 



.^ 



^V, 



g Line of sfretehout 3 



Pattern 

fie. Z9 



°% 



H 



Figs. 26-29.— Candy Pan. 



RECTILINEAR FIGURES 19 

from the side already drawn. A j-incli lap is added to the long 
sides of the pattern at each corner. 

All necessary dimensions should be placed on the plan and 
elevation, and all over-all dimensions on the pattern. 

14. Related Mathematics on Candy Pan. — Problem 3 A. — 
The candy pan shown in Fig. 26 is to be made of IXX Charcoal 
Tin. (Read two cross charcoal tin.) This tin is generally carried 
in stock in two sizes of sheets, 14"X20" and 20"X28". Calcu- 
late the area in square inches of a sheet 20"X28". 

Problem SB. — What is the area of Fig. 29? Use over-all 
dimensions. 

Problem 3C. — What is the largest number of blanks (Fig. 29) 
that could be cut from a sheet of 20"X28" tin? 

Problem 3D. — What are the dimensions of the pieces of tin left 
after cutting the blanks from the sheet? 

Problem 3E. — What is the total area of the pieces of tin left? 

Problem 3F. — Divide the total area of tin wasted (Problem 3E) 
by the total area of the sheet (Problem 3A). The result will be 
the percentage of the 20"X28" that is wasted. 

Problem 3G. — Divide the total area of tin wasted (Problem 3E) 
by the number of blanks obtainable (Problem 3C). This will give 
the amount of tin wasted per blank. Divide this result by the 
total area of one blank (Problem 3B) to get the percentage of waste 
per blank or per pan. 

Illustrative Examples 

Tin blanks 6"X8" are to be cut from a sheet of 14"X20" tin plate. The 
problem is to find the maximum number of blanks obtainable and the per- 
centage of waste. 

Example of Problem 3 A. 
width 14" 

X 
length 20" 

280 sq. iQ., area. Ans. 280 sq. in., area. 

Example of Problem SB. 
width 6" 

X 
length 8" 

48 sq. in., area. Ans. 48 sq. in., area of blank. 

Example of Problem 3C. 

(See Fig. 30.) Ans. 4 blanks. 



20 SHEET METAL DRAFTING 

Exaviple of Problem 3D. 















w 




s 


. 




1 ~1 

f 1 

! i 
1 1 





8" 


8" 


-4' 




Fig. 30. 




Exa77iple of Problem 3E. 




Arts. 2"Xl6"and4"Xl4" 


2" X 16" = 32 




or 


4" X 14" = 56 




2"X20"and4"Xl2" 



Total 88 sq. in. 
Example of Problem, 3F. 

(Area of sheet) 280 



Ans. 88 sq. ill., total waste. 
. 000 (area of waste) | . 314 



84.0 


4 00 
2 80 


1 200 
1 120 



80 



Ans. 31.4%, waste per sheet. 



Example of Problem 3G. 

88 4-4 = 22 sq. in., waste per blank. 
(Area of blank) 48 ! 22 . 00 (waste) | .46 (approx.) 
19 2 



2 80 
2 88 



Ans. 46% waste per blank. 



RECTILINEAR FIGURES 21 

Problem 4 
BREAD PAN 

15. The Iron Bread Pan. — The particular feature of the con- 
struction of this bread pan is the method of joining the body to the 
end by double seams. 

The bread pan, Fig. 31, is to be made of No. 28 black 
iron. It is to be wired with No. 8 wire and the ends are to 
be double seamed in. Figure 36 shows that this double seaming 
is accomplished by turning a hook on the body and a right-angled 
bend on the end. After being slipped together, these edges are 
hammered down to form the double seam. The Double Edge is 
the trade name for the hook that is turned on the body. The 
Single Edge is the trade name given to the right-angled bend on 
the end. Allowance must be made for the metal necessary to 
make these bends. This allowance is called the ''Take-up" and 
is indicated in Fig. 36. 

The end elevation is drawn first and the points of the profile 
numbered 1, 2, 3, and 4 as shown in Fig. 32. The front elevation 
can be located by using the extension lines from the end eleva- 
tion. The line of stretchout is drawn at right angles to the 
bottom of the pan. The spacing of the profile and the corre- 
sponding numbers are then transferred to the line of stretchout. 
This pan is to be wired around the top with a No. 8 wire which is 
■^ in. in diameter. After this allowance for wiring is computed, 
this distance must be added to the line of stretchout outside of 
points 1 and 4. After the measuring lines are drawn on the 
pattern, extension lines are dropped from the front elevation into 
the stretchout. These will locate the extreme points of the 
top and the bottom and permit the drawing of the outline of the 
body pattern as shown in Fig. 34. Three-eighths inch double 
edges are added as shown. The bending lines of the double 
edges are drawn | in. in from the outside edge. This allows -^ in. 
for take-up. 

The pattern of the end is constructed by dropping extension 
lines from the end elevation and by carrying extension lines over 
from the upper surface of the body pattern. The intersections of 
these extension lines will locate the corners of the end pattern as 
in Fig. 35. Three-sixteenths inch edges are added on the three 



22 



SHEET METAL DRAFTING 



j|"Oouble seam 




8 Wire 



Fig. 31 




Figs. 31-36. — Iron Bread Pan. 



RECTILINEAR FIGURES 23 

sides as indicated. The bottom corners of the end pattern are 
notched straight across. The single and double edge notches 
should be dropped below the bending line at the top of the pan a 
distance equal to the diameter of the wire. This will allow the 
wire to lay against the pan instead of riding over the double 
seams. 

All necessary dimensions should be placed on the front and end 
elevations and the over-all dimensions on the pattern. 

16. Related Mathematics on Iron Bread Pan. — Trapezoid. — 
A trapezoid is a flat surface bounded by four straight lines only 
two of which are parallel. The parallel sides are known as the 
upper and lower bases of the trapezoid. 

The area of a trapezoid is equal to one-half the sum of the bases 
multiplied by the altitude. The altitude of any surface is always 
the shortest distance between its upper and lower parts. The 
altitude must always be measured at right angles to the lower 
base. 

Problem J^A. — How many body blanks, Fig. 34, can be cut from 
a sheet of black iron 30" wide and 96" long? Treat the pattern 
as a rectangle using the over-all dimensions. 

Problem 4B. — Could any of the pieces left from the body blanks 
be used for end blanks? If so, how many end blanks could be 
obtained from these? 

Problem 4C. — What is the total area of the waste pieces? 

Problem J^D. — What is the percentage of waste for one body 
blank? 

Problem 4E. — By reversing the end pattern when laying out 
on the sheets some material may be saved. Show by a sketch 
how to effect this saving of material from a 30"X96" sheet. How 
many end blanks can be obtained from one of these sheets? 

Problem 4F. — What is the total area of the waste pieces from 
one of the above sheets? 

Problem 4G- — What is the area of one end blank? What is 
the percentage of waste? 

Problem 4H- — The bread pans are to be made of No. 28 black 
iron weighing .625 lb. per sq. ft. How much will 1000 body 
patterns weigh after corrections are made for waste? How much 
will 2000 end patterns weigh after corrections are made for waste? 
What will be the correct weight of 1000 of these pans? 



24 



SHEET METAL DRAFTING 



Illustrative Examples 

1000 bread pans 4"X8"X2" deep with |" flare on all sides are to be made. 
The size of the body pattern is 7|" by 8|". The size of the end pattern is: 
top 4^", bottom 3^-", depth 2|". 

Example of Problem 4A. 



111 



CO 



:4«J 
CO 



J 



|-7|- 



.96'- 



-Thirty four more 



Eleven more 

Fig. 37. 



JL 



-HilK 



Ans. (a) 36 blanks for body. 
(&) 38 blanks for ends. 
Example of Problem 4C. 

1 piece 1|"X25|" = 38.25 sq. in. 
1 piece 4|" XI" = 4.50 sq. in. 

Total 42.75 sq. in. Ans. 42.75 sq. in., waste from bodies. 

Example of Problem 4D. 
length 8.5" 



X 
width 7.875' 



Total waste 42.75 h-36 = 1.18 sq. 
in., waste per blank. 



66.93 sq. in., area of body. 
(Area) 66.93 1 1 ■ 1800 (waste) | .017 
6693 



51070 
46851 



Ans. 1.7%, waste per body blank. 



RECTILINEAR FIGURES 

Example of Problem 4E. 



25 



—If 

If 






■37 more 



1 .......J 



95" • 

Fig. 38. 

Ans. 266 end blanks. 
Number of blanks from width of sheet 7 
" length " " 38 

Total number of blanks 38X7= 266 

Example of Problem 4F. 

One piece 95"Xl|" = 142i sq. in. 
" " 30" Xl" = 30 " " 

Total waste 172| sq. in. Ans. 172^ sq. in., waste from end. 

Example of Problem 4G. 

The end blank is in the form of a trapezoid, the area of which is equal to 
one-half the sum of the upper and lower bases multiplied by the altitude. In 
this example the lower base is 4|", the upper base 3|", and the altitude 2|". 

42"+3|"= 8", sum of lower and upper bases; 
8" -i-2 = 4", half of the sum of the bases; 
4" X2|" = 10 sq. in., area of trapezoid. 

(No. of end blanks) 266 | 172 . 50 (area of waste) | .64 sq. in., waste per blank. 
159 6 



12 90 
10 64 



(No. of blanks) 266 | 640 (waste per blank) [ .002 
532 
Ans. (a) 10 sq. in. 

(b) To% waste. 



26 SHEET METAL DRAFTING 

Example of Problem 4H. 

Area of one body blank 66.93 sq. in. 
Area of 1000 body blanks 66930 sq. in. 

464 .8 area in sq. ft. (approx.) 
Sq. in. in one sq. ft. 144 | 66930 
576 



933 

864 



690 

576 

1140 
1152 

464.8 area in sq. ft. 

1 .017 correction for 1.7% waste. 



32536 
4648 
46480 



472.7016 corrected area. 

472.70 area in sq. ft. 
. 625 wt. per sq. ft. 



236350 
94540 
273620 



285.43750 lb. wt. of 1000 body blanks. 

Area of one end blank 10 sq. in. 
Area of 2000 end blanks 20,000 sq. in. 
138.88 area in sq. ft. (approx.) 
144 I 20000.00 
144 

560 
432 

1280 
1152 



1280 
1152 



1280 Aws. (o) 285.43 pounds. 
1152 (6) 86.96 pounds. 
(c) 372.39 pounds. 



RECTILINEAR FIGURES 27 

138.88 area 
1 .002 correction for to% waste. 



27776 

1388800 



139.15776 sq. ft. corrected area. 

139.15 area 

. 625 285 . 43 corrected wt. of 1000 bodies 

-— — 86.96 " " of 2000 ends 

69375 , 



27830 372.39 weight of 1000 pans. 

83490 



86.96875 lb., wt. of 2000 end blanks. 



CHAPTER II 
WIRED CYLINDERS 



Prob. 
No. 



Job 



Drana/ I NG 
Obje CTIVE 



NlATHEMATfCAU 

Objective 



<^ 



Chimney 
Cap 



e/ei^af/on of a 

cy hnder. 
/c/ea of e/jve/ope 
of cy///7c/er. 
Sfandarc/ /ocM, 



C/'rc umferen c e 
of a c/'rc /e, 

Lafera/ area 
of cy//'nder. 




/yotc/7/ng docfy 
/:^atfer/7. 

8urr or? i>oTto/v. 



Area of c/rcfe. 



o Pint Cu p 



Z Qt. Painter's Pail 




Doub/e searn/ng 
do ffom. 

W/'re £>a/7. 



Ca. incAes in a 

ga//o/7. 
Compuf/n^ vo/c/me 

of cy//nder. 
Conien'fi '//? porfs 

of a ^a//on. 



Garbage Can(No coirer) 



8 




Beadinff corf. 
Ba'// ear3 a/}c/ 



Co/npiyf/n^ i/aknom 

d/mens/ons, otijer 

d/'/ne/Jsions de/ng 

g/\/en. 



Objectives of Problems on Wired Cylinders. 
29 



30 SHEET METAL DRAFTING 

Problem 5 
GALVANIZED CHIMNEY TUBE 

17. The Chimney Tube. — In the problem of the chimney tube, 
Fig. 39, the student will get a clear understanding of the idea of 
unroUing the envelope of an object to get the pattern. 

A chimney tube is a short piece of pipe intended to be built 
into a chimney. If the tube is held on a level with the eye, the 
sides, the top, and the bottom will appear as four straight lines. 
The elevation of Fig. 40 shows this view. It is impossible to tell 
whether this elevation represents a flat or a curved surface unless 
another view is drawn. For this reason a profile should be drawn. 
This profile will show that the elevation is that of a cylinder. 
Extension lines are used to locate this profile properly. Fig. 41. 
Three dimensions are given in the elevation. The elevation 
shows a f -inch flange on the top end of the tube. It is unnecessary 
to draw this flange in the profile. 

Figure 43 represents a cylinder placed on its side. The profile 
appears on the left-hand end. Straight lines are drawn on the 
body from each division of the profile, parallel to the sides of the 
cyhnder. If each fine left a mark as the cylinder was rolled 
along a flat surface, we would obtain the stretchout as shown. 
The lines running from the top to the bottom are the measuring 
lines of the stretchout, since upon these lines any point on the 
surface of the cylinder can be measured (located). This illustra- 
tion also makes plain the reason for drawing the line of stretchout 
at right angles to the view from which the pattern is to be taken. 

The profile, Fig. 41, is divided and the divisions numbered as 
shown. The fine of stretchout should be drawn and the spacing 
of the profile transferred to this line. The numbers should 
correspond. It should be remembered that it is necessary to 
start with the number of the profile at which the seam is to occur 
in the finished article. Perpendicular fines should be erected at 
points 1 and 1 of the stretchout. Extension fines drawn from the 
elevation complete the stretchout. A |-inch edge is added to allow 
for the flange called for in the elevation. On the right and left 
edges of the stretchout, |-inch edges for locks are added. These 
locks must be turned in the stove-pipe folder. The top of each lock 
is notched to reduce the thickness of the seam on the flanged end. 



WIRED CYLINDERS 



31 




Fig ..3 9 



-nih— 4- 




n^ 



g" Flange 



-Lock 



Line, of stretchout 



I a3-4 567 8 9lOllltl 

Pattern of body 



Lock 



FiG.4i 




Measuring lines 
of stretchout- 



Figs. 39-43. — Galvanized Chimney Tube. 



32 SHEET METAL DRAFTING 

18. Related Mathematics on Chimney Tube. — Circumference 
of a Circle. — A circle is a portion of a flat surface bounded by a 
curved line, every point in which is the same distance from a 
point within, called the center. The circumference is simply 
the curved line that is drawn with the compass. The diameter of 
a circle is any straight line that passes through the center of the 
circle and has its ends in the circumference. It is possible to 
draw any number of diameters in the same circle, but they will 
all have the same length. 

Value of TT. — The girth or distance around a cylinder can be 
found by wrapping a narrow strip of newspaper around it. The 
point where the strip overlaps the end should be marked before 
the paper is removed from the cylinder. The distance from the 
end of the paper to the mark wiU give the distance around the 
cyhnder, or the girth. 

If the diameter of the cylinder be accurately measured and 
the cu'cumference or girth divided by the diameter, the answer 
will be about Sy. Regardless of the size of the cylinder this 
experiment will always produce the same result. Mathematicians 
have proved that the exact relation of circumference to diameter 
cannot be found. The value 3.1416 is near enough for most 
purposes. Some sheet metal workers use 34^" or ^-^ in their com- 
putations. This relation between circumference and diameter is 
indicated by the Greek letter tt (pronounced pi). 

Suppose it is desired to find the circumference of a 7" circle: 
7"X^2__i|A = 22", or?" X3. 1416 = 21. 9912". If the circum- 
ference is given and the diameter is wanted, the process is 
reversed; i. e., with a circumference of 26", 26"-^^^- = 26"Xt2 = 
8tV'; or 26" ^3. 1416 = 8.2442". 

Lateral Area of a Cylinder. — Lateral means pertaining to the 
side. Lateral area is the area of the side wall of a cylinder. The 
pattern of the side wall of a cyhnder is a rectangle whose length is 
equal to the circumference of the profile, and whose height is the 
height of the cylinder. The area of the pattern is equal to the 
length times the height; therefore, the lateral area of any cylinder 
must be equal to the circumference of the base times the height. 

Problem 5 A. — Compute the circumference of the chimney tube, 
Fig. 42, and compare the answer to the length of the line of stretch- 
out between points 1 and 1. These should agree or a mistake has 
been made. 



WIRED CYLINDERS 33 

Problem SB. — Compute the lateral area of the cylinder shown 
in Fig. 42, without the locks and flange. 

Problem 5C. — Compute the area of the pattern with locks and 
flange added, Fig. 42. 

Note. — The lateral area of a cylinder does not include any 
locks, laps, or flanges, and in order to arrive at the cost of material 
these must be added to the lateral area. 



34 SHEET METAL DRAFTING 

Problem 6 

HALF-PINT CUP 

19. The Half-pint Cup.— This problem is intended to bring 
out the method of notching employed when a wire is rolled into a 
cylinder, to describe the standard "tin lock," and to show how a 
bottom is snapped on. 

In drawing the elevation of the half -pint cup, special attention 
should be given to the following items : The lines representing the 
wire must be y^ in. apart. The lines at the bottom must be | in. 
apart. The handle must be drawn according to the dimensions 
given in Fig. 49. The profile is located by dropping extension 
hues from the elevation. At a distance of If in. from the lower line 
of the elevation, the horizontal center line of the profile should 
be drawn. The extension lines dropped from the elevation should 
intersect the center line, thereby setting off the horizontal diameter 
of the profile. The profile is drawn with the compass after the 
center of this diameter is located. The handle of the cup is shown 
attached to the profile, but it is not essential that this be drawn, 
since the pattern of the handle is a regular taper from a width of 
I in. at the top to \ in. at the lower end. 

The profile is divided into equal spaces and each division 
numbered. After the Une of stretchout is drawn, the spacing of the 
profile is transferred to this line and the divisions numbered to cor- 
respond. At the points 1 and 1 of the line of stretchout per- 
pendicular lines are erected. The stretchout is finished by 
extension lines carried over from the elevation. The wire edge 
which must be computed is added to the top edge of the stretch- 
out. A j-inch edge is added to each side for a standard "tin lock." 
Since a lock has three thicknesses the full allowance is never 
turned. For a tin lock -^ in. must be turned in a bar folder. The 
notching of the wire edge in Figs. 50 and 51 never goes in as far as 
the circumference line, and always goes down below the top line \ 
of the stretchout a distance equal to the diameter of the wire. 
This notch removes the thick seam on the body at the point where 
the wire crosses. The bottom of the lock is notched as shown in 
Fig. 51. 

The pattern of the bottom of the cup is drawn by first repro- 
ducing the profile and adding a |-inch edge all around. This edge 



WIRED CYLINDERS 



35 



Fig. 44 



(2. wire 




Wire edae^ 



F«G. 47 



^ ^ V'"^ , o'f stretchout 



Pottern of body (50 wanted) 



T 

ii 



gsingle 
edge 




Pattern of hondle 
(*20 googe) 

FlQ,49, 




r*--No+chinq fo 



^ Allowonce 
y-for lock 



;|^' Turned in 
bor folder 



This is -the 

circumference 

line. 



Notching lower end 
of lock 



Figs. 44-51.— Half-pint Cup. 



36 SHEET METAL DRAFTING 

is turned up in the "thin edge" and is "snapped on" over the lower 
edge of the body. The profile of the handle is shown in elevation. 
This profile is divided into equal spaces. This spacing is trans- 
ferred to any straight line and perpendiculars erected at the 
first and last points. Using the Une of stretchout as a center Hne, 
I in. is set off on each side for the width of the top and | in. on 
each side for the width of the bottom. The pattern of the handle 
is completed by connecting these points with straight lines. The 
handle is intended to be made from No. 20 gage iron, tinned. 
Should the handle be made from Hghter material, it would be 
necessary to add a hem to the long sides of the pattern in order to 
gain the necessary rigidity. 

20. Related Mathematics on Half -pint Cup.— Problem 6A.— 
How many sheets of tin plate measuring 20" X 28" would be re- 
quired to make fifty half-pint cups? Treat the bottom of the cup 
as a square piece of metal. 

Problem 6B. — What would be the percentage of waste for the 
entire job? 

Problem 6C.— 20" X 28" IX "Charcoal Tin, Bright" is packed 
by the manufacturers in boxes containing 112 sheets. If this 
grade of tin plate is selling for $26 per box, how much will the tin 
required for fifty half-pint cups (Problem 6A) cost? 

Area of a Circle. — The method of calculating the area of a 
circle will be thoroughly understood by the student if he will go 
through the following exercise: 

Draw a 5" square. Draw straight lines connecting opposite 
corners of this square. These lines are called the diagonals of the 
square. The diagonals of a square, or rectangle, always divide 
each other into two equal parts. Using the point where the 
diagonals cross each other (intersect) as a center, draw a circle 
that will just touch the center of each side of the square. What is 
the diameter of this circle? How does this diameter compare 
with the length of the sides of the square? You have drawn what is 
known as an inscribed circle; that is, a circle whose circumference 
touches all sides of the containing figure but does not pass beyond 
the sides. What is the area of this square? Would you get the 
same answer if you simply multiplied the diameter by itself? 
This operation is known as "squaring the diameter" and is always 
written D^. Look up a table of areas of circles and you will find 
the area of a 5" circle given as 19.635". Now, divide the 



WIRED CYLINDERS 37 

area of the 5" circle by the area of the 5" square. Is your answer 
.7854? If you should try this experiment with a circle of any 
diameter you would get the same result. Therefore, by squaring 
the diameter of any circle and multiplying by . 7854, you can find 
its area. You will often see this rule written A=D^X .7S54:. 
Does the method of arriving at this result resemble the one em- 
ployed in establishing the rule for finding the circumference of a 
circle? In each case did we divide one quantity by another? 
Dividing one quantity by another establishes a comparison of the 
size of one to the size or the other. This comparison is called a 
ratio. For instance, the ratio of the foot to the inch is 12, and is 
found by dividing the foot by the number of inches in a foot. 
What is the ratio of the yard to the foot? 

Problem 6D. — What is the area of the pattern for the bottom 
of the haK-pint cup, Fig. 48. Compute the area of a 1" circle. 
Compute the area of an 8" circle. Compute the area of a 9|" 
circle. 



38 SHEET METAL DRAFTING 

Problem 7 

PAINTER'S PAIL 

21. The Painter's Pail.— The Painter's Pail, Fig. 52, is gen- 
erally made of No. 28 Black Iron. The bottom of the pail is 
double seamed but it is not soldered. The wire bail is formed with 
a hook on each end. These hooks are inserted in holes punched 
through the sides of the pail. 

A full size elevation, using the dimensions given in Fig. 53, 
should fii^st be drawn and dimensioned. The hnes representing 
the wire at the top of the pail should be slightly more than | in. 
apart. Two lines at the bottom represent the double seam and 
should be ^ in. apart. The upper left-hand corner of the elevation 
should be ''broken" in order to determine accurately the profile 
of the " hook " on the end of the bail. Extension lines drawn down- 
ward from the elevation locate the profile, Fig. 56. The hori- 
zontal center line of the profile should be drawn at a distance of 
three inches from the elevation. By means of the "T-square" 
and triangle a vertical center line of the profile is put in. The 
profile is then completed. The center lines will indicate four 
points on the circumference. These points are to be numbered 
1, 5, 9, and 13 as shown. In order to divide the circumference 
into sixteen equal parts, as indicated, the student should proceed 
as follows: 

With points 1 and 5 as centers, draw two arcs that cross 
each other as at A. You may use any radius in drawing these 
arcs. Carefully connect point A with the center of the profile 
by a straight line. This line will divide that part of the profile 
between points 1 and 5 into two equal parts. Number this 
center point 3. With points 1 and 3 as centers repeat this opera- 
tion, thereby obtaining point 2. The space between points 1 and 
2 may be used to divide the profile into sixteen equal parts. 

The straight line from point A to the center of the profile 
also divides the angle formed by the horizontal and vertical center 
lines into two equgil parts. The angles shown in Fig. 54 are to be 
bisected. Since these angles have no arc shown, it will be neces- 
sary to draw one. The corner (vertex) of the angle should be 
used as a center. The radius should be as large as possible and 
yet have the arc cut the sides of the angle. This arc will give 



WIRED CYLINDERS 



39 



points corresponding to points 1 and 5 of the profile. These 
points are to be used as centers from which to draw the inter- 
secting arcs. 

The line of stretchout, Fig. 55, can now be drawn and the entire 




stretchout developed. A standard tin lock is added to each side 
of the stretchout. Why does the stretchout start with point 1? 
Why do the holes for the bail occur on hnes 5 and 13? A |-inch 
single edge is added to the bottom of the stretchout. The pattern 
for the bottom, Fig. 57, may be drawn by reproducing the profile 



40 SHEET METAL DRAFTING 

and adding a j-inch double edge all around. The double seam on 
this pail is of the same construction as the one shown in Fig. 36. 
The wire edge is added to the top of the stretchout. Using the 
bail shown in the elevation as a profile, the stretchout for the 
wire blank, Fig. 58, can be determined in the usual manner. 

22. Related Mathematics on Painter's Voil.— Volume of a 
Cylinder. — The volume of a cube is equal to the length of the base, 
times the width of the base, times the height of the cube. This 
is written V = LXWXH. It has also been found that length 
times width gives the area. Because of this it can be said that 
volume equals area times height, and that the volume of a cylinder 
is equal to the area of the base times the height. The base of a 
cyhnder is a circle, the area of which equals D^ X . 7854. There- 
fore, for a cyhnder. Volume equals Diameter squared times. 7854 
times the height, or F = D^ X . 7854 X //. 

Sample Problem. — Find the volume of a cylinder 4" in 
diameter and 6" high. 

F = Z>2X.7854X/^ 

F= 42 X. 7854X6 
F=16X. 7854X6 
F = 75 cu. in. Ans. 75 cu. in. 

Problem 7 A. — Compute the volume of the Painter's Pail, 
Figs. 52 and 53. 

Cubic Inches in One Gallon. — It is established by law that one 
gallon of liquid measure shall contain 231 cubic inches. 

Problem 7B. — How many cubic inches are there in one quart? 
In one pint? 

Problem 7C. — What is the exact capacity of the pail. Fig. 53, in 
quarts and fractional parts of a quart? 

Problem 7D. — If a job called for a pail 8" in diameter and 
7|" high, what would be its exact capacity in quarts? 



WIRED CYLINDERS 41 

Problem 8 
GARBAGE CAN 

23. The Garbage Can. — In developing the patterns for large 
objects such as a garbage can, Fig. 59, it is necessary to make the 
drawings to scale. Drawing to scale means making each line on 
the drawing one-half, one-third, one-quarter, or other fractional 
part of its true length. Scale rules are provided to assist the 
draftsman in this work. In order to understand the principles of 
the scale rule, the student should construct a model of one accord- 
ing to the following directions: 

Suppose we wish to make a drawing to a scale of three inches to 
the foot (3" = I'O")- This would call for the use of a rule having a 
three-inch scale. We will proceed to construct such a scale. 
Procure a strip of thin cardboard twelve inches long and one inch 
wide. Lay off along one edge spaces three inches apart. Each 
of these spaces will represent twelve inches on the finished article. 
How many feet of the finished product are represented by the entire 
rule? Mark these divisions 0, 1, 2, and 3. Divide the first space 
into twelve equal parts. Each new space represents one inch. 
Mark the third, sixth, and ninth spaces as shown. Each space 
representing one inch can now be divided into four, eight, or sixteen 
equal parts, in order to represent j in., | in., or 3^ in. respectively. 
The rule shown in Fig. 60 is measuring a distance of three feet, 
four inches (3'-4"). Notice that the feet are read to the right 
of the zero mark and the inches to the left of the zero mark. 

The elevation of the garbage can. Fig. 61, should be drawn 
according to the dimensions shown. This elevation shows two 
0-G beads on the body of the pail. The size of these beads is 
not given, because the equipment of beading rolls varies with 
different shops. The necessary rigidity will be obtained regard- 
less of the size of bead used. Bail ears are used on this job. The 
student's attention is called to the location of these ears. This 
can is designed for a cover that has a rim fitting inside of the body. 
If the cover rim were fitted over the outside, the ears would have 
to be placed below the bead and the bail lengthened to correspond. 
The development of the patterns is similar to the preceding prob- 
lem and needs no additional description. All dimensions on 
the elevation and the patterns should be full size. These are 



42 



SHEET METAL DRAFTING 



known as Witness Dimensions, and the workman always follows 
these while manufacturing from scale drawings. The use of 
witness dimensions relieves the workman from responsibility for 
errors, transferring this responsibility to the draftsman or designer. 




Pattern 
Fig. 60 


—Hi 







fe-n 




P-iO 



24. Related Mathematics on Garbage Can. — Problem 8 A. — 
Compute the exact capacity of the garbage can, Fig. 61, in gallons. 

Transposing a Formula. — Very often a customer requires a 
certain capacity in a vessel but does not care about the dimen- 
sions. In the example of Article 22 the diameter and the height 



WIRED CYLINDERS 43 

were given and the formula V=AXH was used. If the volume 
and the height were given, to find the area the volume would be 
divided by the height. If the volume and the area were given, 
to find the height the volume would be divided by the area. If 
the volume and the diameter were given, the preceding formula 
would be used first, finding the area corresponding to the given 
diameter. 

(a) Volume = Area X Height 
V=AXH 

(h) Area = Volume -f- Height 
A = V-^H 

(c) Height = Volume -^ Area 
H=V^A 

Formula (a) is the original, or basic, formula while (6) and (c) 
are obtained by changing the position of certain quantities. 

What happens to the multiplication sign when area is carried 
over to the left-hand side in place of volume? Does the same 
thing happen when height and volume are interchanged? This 
process of changing the location of the terms in a formula is called 
transposing the formula. When any terms are transposed, the 
sign must also be changed to the opposite; that is, multiplication 
becomes division, addition becomes subtraction, and so on. 

Problem 8B. — What is the area of the bottom of a garbage 
can 16" high, the volume of which is 42 qts.? 

Finding the Diameter of a Circle from the Area. — The formula 
for the area of the base of a cylinder is A = Z)^ X . 7854. Problem 
8B gives the area of the bottom of the can. Before the bottom 
can be made, its diameter must be found. There are two ways 
of doing this: by using printed tables giving this information; or 
by transposing the formula for area and finding the diameter by 
square root. A sheet metal worker must know how to find the 
square root; consequently, the student is advised to become 
famiHar with this process, 

Original formula Area =D^X. 7854 

Transposing, D^ = Area -f- .7854 

(d) Extracting square root, D = VArea -^ . 7854 ♦ 



44 . SHEET METAL DRAFTING 

Problem 8C. — What is the diameter of the can mentioned in 
Problem 8B? 

Illustrative Examples 

Exa?npie of Problem 8B. 

What is the area of the bottom of a can 12" high holding 20 qts.? 

20 qt. = 5 gal. (volume) 

5X231 = 1155 cu. in. (volume) 

Formula (h) would apply here, Area = Volume -f- Height 

Substituting known values, Area = 1155 cu. in. -7-12" 

12 I 1155.00 I 96. 25 sq. in. 

Ans. 96.25 sq. in. 
Example of Problem 8C. 

What is the diameter of the bottom of the can mentioned in the preceding 
example? 

Formula {d) would apply Diameter = VArea -4- . 7854 
Substituting, Diameter = ^96 . 25 -^ . 7854 

■ 7854 I 96.250000 | 122.54 sq. in. 





78 54 




17710 




15708 




20020 




15708 




43120 




39270 




38500 




31416 






Extracting square 


root., v' 122.54 1 11.0 




1 




21 1 022 




21 




220 1 154 



Ans. 11", diameter. 



CHAPTER III 



CYLINDERS CUT BY PLANES 



Prob. 
No. 



Job 



Drawj n g 
Objective 



Math ematical 
Obje ctive 



Scoops 



/dea of cy///?der 
ce/f by i7 />/c7^e. 
Cy///J<:/r/c a/ /?a/7d/e. 



£'sf/'/?7af//7f Cost. 



10 

& 

II 



Conductor Pipe 



n 



Elbows 



M€ff70(:fs o-f 
consfrucf/on 
5tar?e/or<:/ cc/ts 
of /'//^e. 



Ssf/'rr? a /-/'/? ff cat 



\Z 




Afeffjoc/s of 
cons fru c f/'on. 

of p/y?e. 



^st/m a t/n^ weight 
and cost. 



Stove 
Pipe 

EUBOWS 



13 

a. 
14 




Center ///7 e 
rad/'u&. 
T/ie dac/fse-/- 
rneftfod. 



F/^(/r//?^ center 

//'/7e racf/i/s. 

We/yhf of 

e/Jfow. 



Objectives of Problems on Cylinders Cut by Planes. 
45 



46 SHEET METAL DRAFTING 

Problem 9 
THE SCOOP 

25. The Scoop. — Figure 66 shows an ordinary flour or sugar 
scoop. Briefly described, any scoop is a cylinder cut off at an 
angle. A head is soldered in, and a handle is attached to the head. 
Figure 72 shows another type of scoop, the body being cut by a 
curved plane and a cylindrical handle being attached to the head. 

The elevation should be drawn, using the dimensions given in 
Fig. 66. It is not necessary to show the handle in the elevation. 
After the profile. Fig. 67, has been drawn, it should be divided 
into twelve equal spaces. Extension lines from each division of 
the profile should be carried through the elevation, Fig. 68, until 
they meet the miter line. 

Definition of a Miter Line. — The miter line is the line of junc- 
tion between two shapes; these shapes may be alike or unlike. 
The miter line of the scoop is the line of junction between the body 
of the scoop and an imaginary cutting plane. 

The line of stretchout is drawn at right angles to the elevation. 
The spacing of the profile must then be transferred to the line of 
stretchout and numbered to correspond. The measuring lines 
are now drawn in. The extension lines from the profile meet 
the miter line at seven points as shown. From each of the seven 
points of intersection on the miter line a dotted extension line is 
carried over into the stretchout. These extension lines must be 
drawn parallel to the line of stretchout. Starting from point 1 
of the profile, follow the extension line until it meets the miter line, 
and from there follow the dotted line until it meets fines 1 and 1 of 
the stretchout. Small circles are placed where the dotted line 
crosses the measuring fines No. 1 of the stretchout in order to 
mark them definitely. In like manner every point of the profile 
can be located in its proper position in the stretchout. A curved 
line drawn through these points will give the miter cut of the 
pattern. A standard tin lock is added to each side as shown. 
Over-all dimensions should be placed on the pattern as shown. 
The pattern for the head can be obtained by reproducing the pro- 
file and adding a |-inch burr. It is not necessary to allow for the 
dish of the head because it is so slight. 

Up to this point the discussion applies to both types of scoop. 



CYLINDERS CUT BY PLANES 



47 



Figs. 66 and 72. It should be noticed in drawing Fig. 68, that 
points 2 and 12, 3 and 11, 4 and 10, 5 and 9, and 6 and 8 fall on 




the same extension hnes. In view of this fact, many draftsmen 
save tune by drawing a half -profile as shown in the elevation of 
Fig. 73. 



48 SHEET METAL DRAFTING 

The pattern of the handle, Fig. 69, is a straight piece of metal 
f in. wide and 3.1416 in. long. Hems are turned on the long side, 
and a f-inch lap added for joining the ends when the piece is 
"formed up." The handle for the scoop shown in Fig. 73 is 
developed by the same method that was used for the body. 
This is not an exact pattern, due to the double curvature, but 
is near enough for practical purposes on small work. The cap 
for the handle is made by a 1^-inch hollow punch on a lead 
piece. By driving the punch and "punching" into the lead 
piece, a burr is formed on the cap. 



CYLINDERS CUT BY PLANES 49 

Problem 10 
TWO-PIECE ELBOW 

26. The Two-piece Elbow. — A model of a two-piece elbow, 
Fig. 76, can be constructed from a cylindrical piece of wood such 
as a broom handle. The handle should be cut through at an angle 
and the two pieces put together so that they will form an angle 
similar to that shown in the elevation of Fig. 77. It should be 
noticed that the cut portions are not circles but that the section 
is longer in one direction than in the other. The two pieces fit 
together perfectly to form an elbow. 

The following facts concerning all elbows are illustrated in 
Fig. 77. They should be memorized by the student. 

The Base Line. — Every elbow is represented as starting from a 
horizontal hne. This line is called the base line of the elbow. 

Arcs of the Elbow. — Every elbow is made around the arcs of two 
circles. These arcs have the same center. 

Center of the Elbow. — The center of the arcs around which the 
elbow is made is also the center of the elbow. 

Throat of the Elbow. — That part of the elbow drawn around the 
smaller arc is the throat of the elbow, and the arc is the arc of the 
throat. 

Back of the Elbow. — That part of the elbow made around the 
larger arc is the back of the elbow, and the arc is the arc of the 
back. 

Throat Radius. — The throat radius is the distance measured 
along the base Une, from the center of the elbow to the throat. 

Center Line Radius. — The center line radius is the distance, 
measured along the base line, from the center of the elbow to the 
center line of the big end. 

The Backset of an Elbow. — The backset is the amount the back 
rises (sets) above the throat. This vertical distance is indicated 
by the dash line drawn horizontally from the highest point of the 
throat of the big end. 

Number of Backsets. — The first piece of an elbow has one back- 
set, the last piece has one, and every other piece in the elbow has 
two backsets. 

Rule for the Number of Backsets. — The number of backsets is 
equal to the number of pieces in the elbow less one, multiplied by 
two. A four-piece elbow would have (4—1) X2 = 6 backsets. 



50 



SHEET METAL DRAFTING 



Big End of an Elbow.— The big end is the piece that starts at 
the base Hne. Its " cut " is equal to the diameter of the elbow X ir, 
plus the necessary allowance for locks or laps. The big end cut of 
a 7-inch elbow would be (7 Xx) + 1 = 22.991 in. (or 23 in.) 




Small End of an Elbow. — The small end is the last piece of an 
elbow. Its "cut" is found by subtracting seven times the thick- 
ness of the metal used from the big end cut. Thus the small end 



CYLINDERS CUT BY PLANES 51 

cut of a 7-inch elbow made from No. 24 U. S. S. Gage would be 
22. 991 -(.025X7) = 22. 991 -.175 = 22. 816 in. (22i| in.) 

Angle of an Elbow. — The angle of an elbow is a measure of the 
opening formed by two straight lines drawn from the center of the 
elbow to the extremities (ends) of the arc of the back. 

Miter Lines of an Elbow. — The miter lines are the Hnes of junc- 
tion between the pieces of the elbow. All miter Hnes must meet 
at the center of the elbow. 

The student is required to make a drawing similar to Fig. 77 
and to name all the parts defined above. The profile should be 
made about four inches in diameter but the size of the drawing 
is left to the student's discretion. 



52 SHEET METAL DRAFTING 

Problem 11 
TWO-PIECE 60° ELBOW 

27. The Two-piece 60° Elbow. — Figure 79 shows the elevation 
of a two-piece 60° elbow having a throat radius of 3 in. to fit over 
a pipe 4| in. in diameter, and to be made of No. 24 galvanized 
steel. The elevation is started by drawing a base hne 7| in. long. 
The base Hne of an elbow is always equal in length to the sum of the 
diameter of the elbow and the throat radius. Using this base line as 
one side, an angle of 60° must be laid off. A distance equal to the 
throat radius (3 in.) is set off from the vertex (point) of the angle. 
The arcs of the throat and back are drawn, using the vertex of the 
angle as a center. The number of backsets in the elbow is equal to 
(No. of pieces -1)X 2. For this elbow it will be (2-l)X2 = 2 
backsets. The arc of the back is divided into as many equal 
spaces as there are backsets in the elbow; in this case, two 
equal parts. The miter Hne is drawn from the center of the 
elbow through the first division of the arc, above the base line. 
Perpendiculars (lines drawn at right angles) to the base line are 
erected from each end of the diameter of the big end. These per- 
pendiculars must stop at the miter line. Straight lines drawn from 
these intersections to the extremities of the arcs complete the ele- 
vation of the small end. 

After the profile, Fig. 80, is drawn, it should be divided into six- 
teen equal parts, and extension fines carried from each division 
up to the miter line of the elevation. The fine of stretchout. 
Fig. 81, is next drawn. The divisions of the profile are transferred 
to the fine of stretchout and numbered to correspond. Number 1 
of the profile must be so placed that it wiU bring the seam of the 
big end in the throat. The measuring fines of the stretchout are 
drawn and extension fines from the intersections of the miter line 
carried over into the stretchout. Each extension line should be 
traced from its starting point in the profile, up to the miter fine of 
the elevation, and thence to a correspondingly numbered fine of 
the stretchout. A smaU circle marks each intersection thus 
obtained. A curved line drawn through these intersections will 
be the miter cut of the first piece (big end) of the elbow. An 
extension line from the base line of the elbow carried over into the 
stretchout completes the pattern for the big end. Lines 1 and 1 



CYLINDERS CUT BY PLANES 



53 




I 



I 



'-k 



54 SHEET METAL DRAFTING 

of the stretchout can be drawn upwards indefinitely. Since the 
miter cut of both pieces are exactly alike, the pattern of the second 
piece can be constructed above that of the first piece. This will 
bring the seam of the second piece on the back. A distance equal 
to the back of the second piece, as shown in the elevation. Fig. 79, 
should be set off above the miter cut on lines 1 and 1. A horizon- 
tal line connecting these points will complete the pattern for the 
second piece (small end). One-half inch locks are added to each 
side of the stretchout. Notice the notching at the miter cut. 
The big and small end cuts should be computed and placed upon 
the pattern. The small end of every elbow is always cut straight; 
i.e., one half of the deduction for the small end is taken off the 
entire length of the lock on each side of the pattern. No piece 
of a cyHndrical elbow should be tapered, as it adds to the diffi- 
culty of assembling, and is of no advantage when erecting a 
piping system. The direction, "big end minus 7 t " in Fig. 81, 
means the cut of the big end minus seven times the thickness of 
the metal used. Figure 78 shows how a piece of papej' fitted to the 
first piece of an elbow would unroll to produce the pattern. 



CYLINDERS CUT BY PLANES 55 

Problem 12 
FOUR-PIECE 90° ELBOW 

28. The Four-piece 90° Elbow. — In laying out this elbow, 
Fig. 82, an angle of 90° should first be drawn. A distance equal 
to the sum of the throat radius and the diameter of the elbow 
should be laid off upon the horizontal side of this angle. The arcs 
of the throat and the back are then drawn in. A four-piece elbow 
has six backsets. Consequently, the arc of the back should be 
divided into six equal parts. Miter lines are next drawn through 
the first, third, and fifth divisions of the arc of the back, above the 
base line. This gives the first piece of the elbow one backset, the 
second piece two, the third piece two, and the fourth, or last 
piece, one. 

Perpendiculars from the starting point of each arc are erected 
until they meet the first miter line. From these points straight 
lines are drawn so that they just touch the arc at one point and 
continue on until they meet the next miter line. In like manner 
straight lines representing the third piece of the elbow are drawn. 
The elevation is completed by straight lines drawn from the inter- 
section of the third miter line to the ends of the arcs. The length 
of each miter line thus established can be tested with the dividers. 
If all are not of equal length, the drawing is incorrect. The 
elevation of an elbow is always drawn around the outside of 
the arcs. The straight lines of the throat and back are never 
drawn inside of the arcs. Many students make this mistake in the 
elevation and thereby produce an elbow wholly different from the 
one intended. The profile, Fig. 83, should be drawn and divided 
into sixteen equal parts. Extension lines are carried upwards 
from each division until they meet the miter line. The line of 
stretchout and the measuring lines of the stretchout. Fig. 84, 
are drawn. The spacings of the profile are transferred to the line 
of stretchout with numbers to correspond. Extension lines from 
each intersection of the miter line are carried over into the stretch- 
out. Each division of the profile should be traced by means of the 
extension lines, first to the miter line, and thence to the correspond- 
ingly numbered line in the stretchout. Each point thus located 
in the stretchout should be marked with small circles. A curve 
drawn through these points will give the miter cut of the big end. 



56 



SHEET METAL DRAFTING 



It has already been shown that all miter cuts in the same elbow 
are exactly alike as to shape and size. Therefore, it is only neces- 
sary to reverse the pattern of the big end to get the miter cuts for 
the other pieces. Figure 84 shows all four pieces as they would 
appear when laid out on the metal in the shop. The man in the shop 
cuts a rectangular piece of iron the proper size, sets off the climen- 



Smalu Eno 




TT 



Fig. 85 



Method of joining 
pieces of elbow 



Figs. 82-85.— Four-piece 90° Elbow. 



sions of the backs and throats, and by reversing the pattern for the 
big end, gets the entire layout. The section in the lower right- 
hand corner. Fig. 85, shows the method used to join the pieces of 
the elbow together. An elbow made in this manner is known to 
the trade as a "peened elbow." The single and double edges are 
prepared in the turning machine (thick edge) and, after being 
slipped together, the double edge is chnched over the single edge 
with the peen of the hammer. 



CYLINDERS CUT BY PLANES 57 

Problem 13 
LONG RADIUS ELBOWS 

29. The Long Radius Elbows. — The principles of pattern 
cutting that apply to long radius elbows such as used in conveyor 
systems are the same as in the preceding problems. There are, 
however, certain rules that apply to ''Blow Pipe Elbows" that 
should be thoroughly understood. Figure 86 shows a partial eleva- 
tion of a five-piece elbow. Since the pattern for all pieces can be 
laid out from the pattern of the first piece, it follows that all 
necessary information can be obtained from an elevation of the 
first two pieces of an elbow. A draftsman rarely draws more than 
two pieces of the elevation and divides the arc of the throat 
instead of the arc of the back. He uses the arc of the throat be- 
cause it requires less room than the arc of the back and pro- 
duces the same result. 

Center Line Radius. — It has been determined by careful experi- 
ment that an elbow having a center line radius equal to twice the 
diameter of the pipe to which the elbow is to be joined, offers the 
least resistance to the flow of air, or other material, through the 
pipe. According to this rule an elbow for 12-inch pipe would have a 
throat radius of 18 in. and a center line radius of 24 in. A blow pipe 
elbow should never be ''peened. " All laps and edges should be 
closely riveted and soldered air-tight, the inside to be made as 
smooth as possible. All laps should be made in the direction of 
flow of air or other material through the pipe. 

Laps for Riveting. — In the case of the "peened" elbow no allow- 
ance is made for joining the pieces. This alters the throat radius 
somewhat but this fact is generally neglected. In blow pipe 
systems the work must be exactly to measurements. Laps for 
riveting are, therefore, added as shown in Fig. 89. It should be 
noted that the rivet holes for the longitudinal seams are on the 
circumference lines of the pattern, while those for the transverse 
seams are in the center of the lap. The rivet holes are equally 
spaced and as the lap is f in. wide, the centers of the holes are f in. 
in from the edge of the lap, and f in. in from the miter cut of the 
adjoining piece of the elbow. Laps are added to one miter cut 
only (of each piece) and start with the lap on the big end. 

Thickness of Metal Used. — Another rule always to be observed 
is to make the elbow at least two gages heavier than the pipe to 



53 



SHEET METAL DRAFTING 



which the elbow is to be joined. The patterns shown in Fig. 88 
should be separated sufficiently to allow for a lap between each 
piece and must be so drawn by the student. The laps should be 
f in. wide. Rivet holes on all sides of each piece should be shown. 
Figure 90 shows the first and second pieces of an elbow after being 
''fitted." The throat is "laid off" with the stretching hammer, 
and the back is "drawn in" with a mallet or raising hammer. By 




fiG 87 -^- 



FiGS. 86-90.— Long Radius Elbow.;. 

this method the miter cuts of each piece are "butted" and the 
true curvature of the elbow preserved. 

30. Related Mathematics on Elbows. — Problem ISA. — Bach- 
sets of an Elbow. — (See description of Fig. 77 for rule.) 

(a) How many backsets has a four-piece, 90° elbow? 
(6) How many backsets has a six-piece, 75° elbow? 

(c) How many pieces has an elbow having fourteen back- 

sets? 

(d) How many miter- lines has a six-piece elbow? 

Problems on the Rise of the Miter Line. — An elbow is always 



CYLINDERS CUT BY PLANES 59 

measured by the degrees of the angle formed by straight lines 
drawn from its extremities to the center of the elbow, Fig. 77. 
Since all backsets in the same elbow are equal, the value of the 
backset can be expressed in degrees. A five-piece elbow has 8 
backsets. A five-piece 90° elbow would have each backset equal 
to 90° ^8 or llj°. The first piece of any elbow contains one 
backset, the last piece one, and every other piece contains two. 
Therefore, the rise of the miter lines for a five-piece 90° elbow 
would be: 

Rise of 1st miter line would be ll|°Xl =lli°, having but one backset. 

" 2nd " " lirx(l+2)=33r, by adding two backsets. 

" 3rd " " lirX(3+2) =561°, by adding two backsets. 

" 4th " " lirx(5+2) =781°, by adding two backsets. 

Problem 13B. — Give the rise of each miter fine in a four-inch, 
four-piece, 75° elbow. 

Problem 13C. — Give the rise of each miter line in a three-piece, 
90° elbow. 

Problem 13D. — Give the value in degrees of the backset of a 
two-piece, 12° elbow. 

Problem 13E. — Give the value in degrees of the backset of a 
three-piece, 24° elbow. 

Problem 13F. — Give the value in degrees of the backset of a 
four-piece, 36° elbow. 

Proble?n 13G. — Give the value in degrees of the backset of a 
five-piece, 48° elbow. 

Problem 13H. — Give the value in degrees of the backset of a 
six-piece, 60° elbow. 

Problem 131. — Give the value in degrees of the backset of a 
seven-piece, 72° elbow. 

Problem 13 J. — Give the value in degrees of the backset of an 
eight-piece, 84° elbow. 

Problem 13K. — For the same big end diameter, why would one 
pattern answer for all of the elbows mentioned in problems 13D 
to 13J inclusive? 

PROBLEMS ON THE «*CUTS OF AN ELBOW" 

Problem 13L. — What would be the " big end cuts " of the follow- 
ing sizes of elbows? Add the standard stovepipe lock of one 
inch. 



60 



SHEET METAL DRAFTING 



(a) An elbow for 12" pipe? 
(6) An elbow for 14" pipe? 

(c) An elbow for 18" pipe? 

(d) An elbow for 24" pipe? 

Problem 13M. — The "small end cut" of an elbow, or pipe, is 
always found by deducting seven times the thickness of the metal 
used from the cut of the big end. What would be the "small end 
cut" of the elbows in Problem 13L, if No. 20 U. S. S. Gage steel 
was used? Number 20 gage is . 037". 

Problem 13N. — Fill in the columns in the table of deductions 
given below. The figures for the third column are obtained by 
multiplying those of the second column by 7. The figures for the 
fourth column are obtained by dividing those of the third column 
by .0156. This will give answers in 64ths of an inch since .0156 
is the decimal for ^. 

Example of columns filled in: 



Gage 


Decimal Thickness 


Decimal Deduction 


Fractional 
Deduction 


No. 23 


.028125" 


. 196875" 


if" (nearly) 



Table of Deductions for Small End Cuts 



u. 


S. S. Gage 


Decimal Equiva- 
lent, Thickness 
in Inches. 


Decimal Deduc- 
tion, 
Thickness X 7 


Fractional De- 
duction, Decimal 
Deduction ^.0156 




No. 










28 


.015625 








26 


.01875 








24 


.025 








22 


.03125 








20 


.037 








18 


.05 








16 


.0625 








14 


.078125 








12 


. 109375 







CYLINDERS CUT BY PLANES 



61 



STANDARD CUTS OF PIPE 

Manufacturers of pipe and elbows have adopted the following 
standards for big end cuts for stove and conductor pipe. 

Table of Standard Big End Cuts for Pipe and Elbows 



Pipe Size 


Stovepipe (1" lock) 


Conductor Pipe (^" lock) 






Size 


Cut 


4" 


14" 


2" 


61" 


41" 


151" 


2|" 


81" 


5" 


17" 


3" 


91" 


51" 


18§" 






6" 


20" 






7" 


23" 






8" 


26" 






9" 


291" 






10" 


32" 







CENTER LINE RADIUS 

As explained in Fig. 77 the center line radius is the distance 
measured along the base line from the center of the elbow to the 
center point of the diameter of the big end. The arc of the center 
Hne is also shown in Fig. 77. 

Problem 13 0. — What will be the center line radius for the 
following elbows? (a) A 14" diameter elbow? (6) A 7" diame- 
ter elbow? (c) A 9" diameter elbow? 



WEIGHT OF AN ELBOW 

To get the weight of an elbow multiply the length of the center 
line arc by the "cut" of the big end, and this quantity by the 
weight per square foot of the material used. 

Example. — What will be the weight of a 4", four-piece, 60° 
elbow made from No. 24 galvanized iron? Diameter of the elbow 
is 4"; Center Line Radius is 8". 

Since the center line radius is 8" the center line arc must be a 
part of the circumference of a circle whose diameter is 16". The 
circumference of a 16" circle =16 X tt or 50.625". Since the 
elbow has an angle of 60° the center line arc can be but -^^-q or ^ of 
the whole circle. Therefore, the length of the center line arc 
would be i of 50.625" or 8 . 377". The big end cut for a 4" elbow 



62 



SHEET METAL DRAFTING 



is 14" and the surface area is 14X8.377=117 sq. in. (nearly). 
One square foot or 144 sq. in. of No. 24 galvanized iron weighs 
1.156 lb. Therefore, 117 sq. in. would equal |^ of 1 . 156 or 
1 . 136 lb. {Ans.) 

The table given below shows the weight in pounds per square 
foot of the gages of metal in common use in the shop. 

Table of Weights Per Square Foot of Galvanized and Black 

Sheets 



Gage 


Galvanized Steel 


Black Steel 


U. S. S. Standard 


Wt. per sq. ft. in lb. 


Wt. per sq. ft. in lb. 


28 


0.7812 


0.6375 


26 


0.9062 


0.765 


24 


1.156 


1.02 


22 


1.406 


1.275 


20 


1.656 


1.53 


18 


2.156 


2.04 


16 


2.656 


2.55 


14 


3.281 


3.187 


12 


4.531 


4.462 



Problem 13P. — How much would a 90°, 7" elbow weigh if 
made from No. 22 galvanized iron? Elbow to have standard 
radius but to be "peened." 

Problem 13Q.— What would be the weight of a 90°, 8" elbow 
having a throat radius of 8" and made of No. 20 black iron? 

Note. — When the throat radius is less than standard, add one- 
half of the diameter of the big end to get the center line radius. 

Problem 13R. — How large a piece of iron would be required to 
make a 75°, 10" diameter, standard blow pipe elbow of eight 
pieces? Add |", for a lap on each piece, to the length of the center 
line radius. 



CYLINDERS CUT BY PLANES 63 

Problem 14 
THE BACKSET METHOD 

31. The Backset Method. — The Backset Method is a short, 
but accurate, method of developing an elbow pattern. Figures 
91 and 92 show the elevation and profile of an elbow, the pattern 
of which is developed in the manner previously described. It 
should be noticed that the pattern has been moved over to the 
right to allow a half circle to be drawn between it and the elevation. 
The extension lines cut this half circle at points A, B, C, D, E, 
F, G, H, and J. These views should be carefully drawn and 
placed in the position shown. 

The distances between A and B, B and C, etc., should be 
exactly equal if the drawing is carefully made. The diameter of 
this half circle is equal to the height of the backset. Because of 
the foregoing, the elevation and profile are not necessary if the 
backset height in inches is known. The half circle divided into 
equal spaces can be used instead. 

Figure 94 shows a five-piece, 90° long radius elbow laid out 
by this Backset Method. A piece of metal is cut of sufficient 
size to make the whole elbow. A horizontal line is drawn 
to represent the height in the throat of the first piece. Above 
this, another line is drawn to represent the height of the backset 
of the elbow. Lock lines are next drawn | in. in from the right and 
left-hand edges of the blank. The half circle is next drawn in the 
backset as shown. This half circle is divided into eight equal 
parts. The girth (distance between the lock lines) is next 
divided into sixteen equal parts. The girth in Fig. 94 is 28 in. and 
each space would equal 28-^ 16 or If in. The dividers can be set 
If in. and the spacing performed without repeated trials. Meas- 
uring lines are then drawn through each division at right angles 
to the bottom of the blank. A rectangular piece of metal with 
one edge turned to a right angle is a convenient tool for draw- 
ing these lines. Extension lines are carried over from each divi- 
sion of the half circle, and points of intersection determined. A 
curved line through these intersections wiU complete the pattern 
of the first piece. Experienced men can locate the intersections 
with four settings of the compass because point B is the same 
distance from the top as point H is from the bottom line. This 



64 



SHEET METAL DRAFTING 



is also true of points E and G, and D and F, while point E is in the 
center. The student should try and see if he can learn the 



Fig. 91 




Figs. 91-94.— Use of the Backset Method. 



method of doing this, thereby saving time by doing away with 
the extension lines drawn from the half circle. The laps neces- 
sary for riveting the pieces together are shown in Fig. 94. 



CHAPTER IV 
INTERSECTING CYLINDERS 



Prob 

No. 



Job 



DRAW I tslG 
Oe>J ECTIVE 



Mathematical 
Objective. 



15 
(3c 
16 




/<5<?<5^ ofc////7- 
c/er cc/f l>y 



/4rec^ ofc/rc/e 



Tee-jo/nts 
ft ri^h^ Qr\q\e5. 



17 
18 




arc /recessary 
for ff?/5 type 
of Tee. 



/^/Ter//7g 7^6> 
size- of /na/'/7 
/v/?e fo co/npen- 
3a te for^ranc/} 
/J/'pes. 



Teejoints 
notdt ri^titdn^les. 



Objectives of Problems on Intersecting Cylinders. 
65 



66 SHEET METAL DRAFTING 

Problem 15 
TEE JOINT AT RIGHT ANGLES 

32. The Tee Joint at Right Angles. — The name commonly 
applied to two pipes that intersect is "Tee Joint." Some forms of 
these joints are known in various localities as "Y-B ranches" and 
"Tee Y's." One of the scoops, Fig. 72, was a cylinder cut by a 
curved plane. A study of Fig. 95 will reveal a similar condition, 
the branch pipe being cut by the curved surface of the main pipe. 

Side Elevation. — The side elevation should be drawn according 
to the dimensions given in Fig. 96. The profile should be drawn 
above the side elevation to avoid a confusion of lines. The profile 
must be divided into sixteen equal parts, and each part numbered, 
Fig. 96. Extension lines are dropped downward from each divi- 
sion of the profile until they intersect the circumference line of 
the main pipe. These intersections, A, B, C, etc., are lettered as 
shown. 

Front Elevation. — Extension lines are used to locate properly 
the front elevation. Fig. 97. A profile is drawn above the front 
elevation. This profile is divided into as many equal spaces as 
there are in the profile first drawn. Both profiles must be of the 
same size since they represent the same pipe. The profile. Fig. 
97, should be numbered as shown. It should be noticed that 
number 1 of the profile of Fig. 96 is at the bottom, while number 
1 of the profile of Fig. 97 is on the left-hand side of the horizontal 
diameter. This is an important fact, and is true of every drawing 
that has two elevations. Whatever is directly in front in the side 
elevation appears on the left-hand side in the front elevation. 
The true shape of the miter line in the front elevation cannot be 
shown until it is developed. To do this, extension lines are carried 
over into the front elevation from points A, B, C, etc., of the 
apparent miter line. Fig. 96. Extension lines are also dropped 
from each division of the profile in Fig. 97. Starting at point 1 
of the side elevation profile, the extension line is traced downward 
to the apparent miter line, and then horizontally to a correspond- 
ingly numbered line dropped from the profile of Fig. 97. This 
point is marked with a small circle. All other points of the profile 
of Fig. 96 should be traced in like manner. A curved line drawn 
through these intersections will give the developed miter line. 



INTERSECTING CYLINDERS 



67 



Pattern for the Tee. — After drawing the line of stretchout, 
the spacing of either profile (both profiles are the same size) is 




-< 


i H 


2? 

<S J 


0) 
01 
O 

u. 

B^ 

"S' 

tr- 

z 

"i 

-r- 

9 




3 


7 

e 


> 



inoi(349^is >a sun 



^ 



rt 



J. 

to 

o 



transferred and numbered to correspond. The measuring lines 
of the stretchout should also be drawn in. Starting at point 1 of 
the front elevation profile, the extension line should be followed to 



6& SHEET METAL DRAFTING 

the developed miter line, and thence to the correspondingly 
numbered line of the stretchout. This intersection is marked with 
a small circle. In like manner all the remaining points in the 
profile of Fig. 97 should be traced and the intersections of each with 
its corresponding measuring line in the stretchout determined. 
A curve traced through these intersections will give the miter cut 
of the pattern. 

Operiing in Main Pipe. — A line of stretchout, Fig. 98, is first 
drawn at right angles to the main pipe. The spacings of the 
apparent miter line, Fig. 97, are set off upon this line, and these 
divisions lettered to correspond to the lettering of the apparent 
miter line. Measuring lines from each point are drawn at right 
angles to the fine of stretchout. The intersections of the developed 
miter line, as shown in Fig. 97, should now be lettered. It 
should be noticed that the position of the lettering is changed as 
was the numbering of the profile, and for the same reason. A line 
is dropped from point E of the developed miter line until it inter- 
sects line E of the stretchout in Fig. 99. The intersection is 
marked with a small circle. In like manner intersections for all 
other points of the developed miter line should be located. A 
curve drawn through these intersections will give the true shape of 
the opening in the main pipe, Fig. 99. 



INTERSECTING CYLINDERS 69 

Problem 16 

TANGENT TEE AT RIGHT ANGLES 

33. The Tangent Tee at Right Angles. — When a straight line 
and the circumference of a circle touch each other at but one point, 
the straight line is said to be tangent to the circle. Their point 
of meeting is called the point of tangency. A straight line from 
the center of the circle, meeting the tangent at an angle of 90°, 
locates the point of tangency. In Fig. 101, the tangent and the 
point of tangency are plainly indicated. 

Tangent Tee. — When the tee joint is so placed upon the main 
pipe that one side of the tee is tangent to one side of the main, as 
in Fig. 100, the fitting is known as a ''Tangent Tee." The use 
of this type of fitting enables the designer to keep the distance 
from the ceihng to the top of every horizontal pipe uniform 
throughout the entire system. All hangers for the piping can be 
made the same length and the entire job will have a neater and 
more workmanlike appearance. 

Side Elevation. — A side elevation should be drawn according 
to the dimensions given in Fig. 101. After drawing a profile, it 
should be divided into sixteen equal spaces, and extension lines 
from each division carried to the apparent miter line. The 
divisions of the profile should be numbered, and the intersections 
of the apparent miter hne lettered as shown in Fig. 101. 

Front Elevation. — The front elevation can be located by exten- 
sion hnes. Another profile should be drawn above the front 
elevation. It must be divided into the same number of equal 
spaces as the profile of Fig. 101. It is to be numbered, placing 
number 1 on the left-hand side. Extension lines are dropped 
from each division of the profile, Fig. 102. Other extension lines 
are carried over from the intersections of the apparent miter 
line. Fig. 101, until they intersect the extension Hnes drawn 
from the profile. Starting from point 1 in profile of Fig. 101, 
the extension Hne should be traced downward to the apparent 
miter line, and thence to a correspondingly numbered Hne dropped 
from the profile of Fig. 102. In like manner, ah the divisions of 
the profile, Fig. 101, can be traced and each intersection marked 
with a smaU circle as shown in Fig. 102. A curved line passing 
through these points will give the developed miter line. One- 



70 



SHEET METAL DRAFTING 



half of the developed miter line is dotted to represent that part of 
the line that cannot be seeii from the front. 




rt 



IQ) sTj -l aui-j j.«»Bur^ 3ijf SI si.(i. 



Si.rj. ^o ^.urofl »^^ 



\^ 



Pattern for the " Tee."— Draw the Une of stretchout, Fig. 103. 
The spacing of the profile should be transferred and the divisions 
numbered to correspond to the numbers of the profile. The 



INTERSECTING CYLINDERS 71 

measuring lines of the stretchout should be drawn. Starting at 
point 1 of the front elevation profile, the extension line is to be 
followed to the developed miter line, and thence to a correspond- 
ingly numbered measuring line in the stretchout. This inter- 
section should be marked with a small circle. All the other 
points in the profile of Fig. 102 are traced in like manner, and the 
intersections for each measuring line in the stretchout determined. 
A curve traced through these intersections will give the miter 
cut of the pattern. 

Opening in Main Pipe. — A hne of stretchout, Fig. 104, must 
be drawn at right angles to the main pipe. The spacing of the 
apparent miter line is to be set off upon this line. These divisions 
are lettered to correspond to those of the apparent miter line. Fig. 
101. Measuring lines, at right angles to the line of stretchout, 
are drawn from each point. The intersections of the developed 
miter are lettered as shown in Fig. 102. The position of the 
letters must be changed in the same way as the numbers in the 
profiles of Figs. 96 and 97. A line should be dropped from the 
point E of the developed miter line until it intersects the line E 
of the stretchout in Fig. 104. The intersection of the two lines 
is marked with a small circle. In like manner, intersections for 
each division of the developed miter line, as shown in Fig. 102, 
can be located. A curve traced through these intersections 
will give the true shape of the opening in the main pipe. 



72 SHEET METAL DRAFTING 

Problem 17 
TEE JOINT NOT AT RIGHT ANGLES 

34. The Tee Joint Not at Right Angles.— The tee joint not 
at right angles is used for conveyor systems, and in heating and 
ventilating work. The abrupt change of direction in the Tee 
Joint at right angles causes a drop in velocity that seriously 
affects the working of an entire system. The greatest angle of 
deflection allowable in a Tee Joint not at right angles is 45° from 
the direction of flow as in Fig. 105. Many firms use an angle 
of 30°. 

Side Elevation. — The side elevation should be drawn using the 
dimensions given in Fig. 106. The branch pipe is represented as 
being broken in this view, because the upper part of the branch 
plays no part in developing the pattern. A profile, Fig. 106, 
divided into sixteen equal parts, which are numbered, is now 
drawn. Extension fines are carried downward from each division 
until they meet the apparent miter line. The intersections of the 
apparent miter line A, B,C, etc., are lettered as shown in Fig. 106. 

Front Elevation. — The front elevation. Fig. 107, should be 
drawn in outline. The "tee" is set at an angle of 45° to the main 
pipe as shown. The profile of the "tee" should be drawn and 
divided into sixteen equal parts. An extension line from each 
division of this profile is carried downward and to the right; that 
is, parallel to the sides of the branch. 

Developing the Miter Line. — Horizontal extension lines from 
each intersection of the apparent miter line are carried over into 
Fig. 107. If a view were to be taken along these lines in the direc- 
tion of the arrow, the eye would see two points, D and F for in- 
stance, on each line. But Fig. 106, the side elevation, represents 
such a view; therefore the intersections of the front half of the 
apparent miter line must each have two letters. Starting at the 
point A in Fig. 106, the intersections of the apparent miter line 
must be lettered as shown. Starting from point 1 in the profile 
of Fig. 106 the extension line can be traced downward to the 
apparent miter line, and thence to the correspondingly numbered 
Une, dropped from the profile of Fig. 107. In like manner, the 
other intersections for the developed miter line are located and the 
curve drawn. 



INTERSECTING CYLINDERS 



73 



Pattern for the "Tee." — After the line of stretchout is drawn at 
right angles to the tee in Fig. 108, the spacing of the profile should 
be set off on it and the divisions numbered to correspond. The 
measuring lines of the stretchout are put in at right angles to the 
line of stretchout. Since the branch is at an angle of 45°, all of this 
construction work can be drawn with the T-square and the 45" 
triangle. Extension lines parallel to the line of stretchout are 




Figs. 105-109.— Tee Joint Not at Right Angles. 



carried from each intersection of the developed miter line until 
they cut corresponding measuring lines in the stretchout. A 
curved line through these intersections will give the miter cut of 
the pattern. 

Opening in Main Pipe. — A line of stretchout, Fig. 109, is drawn 
at right angles to the main pipe of Fig. 107. The spacings of the 
apparent miter line are set off on this line and the divisions lettered 
to correspond. Measuring lines are drawn from each point at 



74 SHEET METAL DRAFTING 

right angles to the Hne of stretchout. The intersections of the 
developed miter hne should be lettered as shown in Fig. 107. An 
extension line is dropped from point E of the developed miter line, 
until it meets line E of the stretchout. In hke manner, inter- 
sections for all other points in the developed miter hne are located. 
A curve drawn through these points will give the true shape of the 
opening in the main pipe. 



INTERSECTING CYLINDERS 75 

Problem 18 
TANGENT TEE NOT AT RIGHT ANGLES 

35. The Tangent Tee Not at Right Angles. — Figure 111 shows a 
side elevation which has the same appearance as the side elevation 
of Fig. 106. The branch pipe, however, is tipped towards the eye 
as will be seen by studying Fig. 112. Every tangent tee at other 
than right angles must have the entire miter line developed. The 
method of drawing this problem does not vary from that of 
the preceding one. Consequently, the method need not be 
repeated here. The student is cautioned to follow each step 
carefully. 

The student has perhaps noticed a similarity of method for 
developing the patterns for ^1 cylinders. Such pattern problems 
come under the head of ''Parallel Line Drawing," which takes its 
name from the fact that the sets of extension and construction 
lines are parallel to one another. The following general rules 
apply to parallel line developments, and, if carefully followed, can 
be applied to any problem of this class with success. 

Rule 1. — Draw a side elevation, if necessary, to show a true 
miter line. 

Rule 2. — Draw a front elevation, if necessary, to show a true 
miter line. 

Rule 3. — Draw necessary profiles. 

Rule 4- — Divide the profiles into equal spaces, and number the 
divisions. 

Rule 5. — Carry extension lines from each division of the profile 
to the miter line. 

Rule 6. — Develop a miter line if necessary. 

Rule 7. — Draw a line of stretchout, transfer the spacing of the 
profile to this line, and number to correspond. 

Rule 8. — Draw the measuring lines of the stretchout. 

Rule 9. — Carry the extension lines over into the stretchout, 
from each division of the true miter line. 

Rule 10. — Trace the intersections of the stretchout, beginning 
at the profile, thence to the miter line, and from there to a cor- 
respondingly numbered line in the stretchout. 

Front elevations. Figs. 107 and 112, could be dispensed within 
tee joints at right angles. As a matter of fact, experienced layer- 



76 



SHEET METAL DRAFTING 



outs never draw them. They are included here for instructional 
purposes. 

36. Related Mathematics on Tangent Tees and Tee Joints. — 

Altering the Main Pipe Size. — In a blow-through system, whenever 
a branch is taken from the main pipe, the diameter of the main 
must be reduced beyond that point. Also, if a branch pipe is 
added on to the main of an exhaust system, the diameter of the 




Figs. 110-114.— Tangent Tee Not at Right Angles. 



main must be increased. By so doing, the original velocity and 
static pressures are preserved and the system exerts an equal 
''pull" at all points. 

Rule for Altering the Size of Alain. — The area of the main must 
be increased, or diminished, by an amount equal to the square 
inches of cross-sectional area of the branch. In other words, the 
cross-sectional area of the main must at all times be equal to the 
combined cross-sectional area of all its branches. 



INTERSECTING CYLINDERS 77 

Sample Problem. — Two heaters are to be connected to a chim- 
ney flue by one pipe. One heater has a 7-inch, and the other 
a 9-inch neck. How large must a main pipe be to care for both 
heaters? 

Formula for area, A =2)2 X . 7854 
Substituting, A= 72X.7854 

= 38.485 sq. in., area of 7" pipe 
and A = 92 X. 7854 

= 63.617 sq. in., area of 9" pipe. 

Since this is an exhaust system the areas must be added to 
get the equivalent area of the main. 

Equivalent area = 38.485+63.617 = 102. 102 sq. in. 

Transposing the above formula, 

i) = VA-^.7854 (Problem 8, Article 24.) 
Substituting in this formula, 

i) = Vl02. 102 ^.7854 

.7854 I 102.1020 1 130 
78 54 



23 562 
23 562 



Vl30=11.4" II A" Ans. 

In actual practice we would make the pipe 11§" in diameter. 

Problem 18 A. — Two heaters are to be connected to a chimney 
flue by one pipe. One heater has an 8-inch, and the other a 
10-inch neck. How large must the main pipe be to serve both 
heaters? 

Problem 18 B. — A battery of three steam heaters having 8|" 
smoke necks are to be connected to the chimney by one main pipe. 
(a) What will be the diameter of the main between the second 
and the third heaters? (6) What will be the diameter of the main 
between the third heater and the chimney? 

Problem 18C. — Six blacksmith forges are to be connected to 
one smokestack. Each forge has a 6-inch neck. Give the size 
for the main pipe as each forge is ''picked up." 

Problem 18D. — In a shavings removal system a "sticker" is 



78 SHEET METAL DRAFTING 

to be provided for. Two hoods on the ''sticker" have 5-inch 
necks, and two other hoods have 4-inch necks. How many- 
square inches should be added to the area of the main pipe to care 
for this machine? 

^Problem 18E. — A forced draft heating system in a factory has a 
9" diameter outlet every 20 feet. The main pipe as it leaves 
the fan is 20" in diameter, (a) How many 9" branch outlets 
wdll this main serve? (6) What will be the diameters of the main 
after each branch is taken off? 

Problem 18F. — Two boilers are set side by side. Each boiler 
has a rectangular smoke neck measuring 14"X37". How large 
must the round pipe be made that is to convey the gases from these 
boilers to the stack? 



CHAPTER V 
CONES OF REVOLUTION 



Prob. 
No. 



Job 



Draw i nj g 
Objective 



Mathematical 
Objective 



19 




Conical Flower Holoeh 



e/fye/o/?e of o 

COfJG . 

£/e/77e/its of a 
si/rface. 



area of & c^o/7&. 



ZO 




co/?strc/ct/on of cover 






Garbage Can Cover 



Cortes cc/f ^y 



Z\ 



C 



Vegetable parer 



Pfaw//7g to 



fr£/sr(/m of a 
cc?/fe. 



23 




£xfe/7s/o/? of /Wee/ 
of co/jes cot l>y 
f/^/pes 

Starjdafd 
CO/73 tri/ction^ 



of f^'/i/sijed arf/c/e 
fram a scafe 
draWf/?£. 



CONICAU 
OOF FLANGE 



Objectives of Problems on Cones of Revolution. 
79 



80 SHEET METAL DRAFTING 

Problem 19 
CONICAL FLOWER HOLDER 

37. The Conical Flower Holder.— The sketch, Fig. 115, shows a 
flower holder, such as is often carried in stock by florists. The 
body of this holder is a right cone. 

Solid of Revolution. — Any plane surface rotating about a fixed 
point, or a line, generates a solid. For instance, a rectangle rota- 
ting about one of its sides generates a cylinder. A right-angled 
triangle rotating about its altitude generates a cone. Such a cone 
is known as a Right Cone or a cone of revolution. 

Axis of the Cone. — The line about which the generating surface 
revolves in forming a solid of revolution is called the axis. It is 
the shortest distance between the apex (point) and the base, and 
forms a right angle to the plane of the base. 

Elevation of the Cone. — The elevation of the cone, Fig. 116, is 
drawn in the following manner: Draw a horizontal line four 
inches long; from the center of this line drop a perpendicular 
seven inches long. Connect the ends of the four-inch line to the 
end of the seven-inch line by straight lines. The four-inch line 
is the Base of the Elevation. The seven-inch line is the Altitude 
of the Cone. The straight lines connecting the ends of the base and 
the altitude are the Slant Height lines of the elevation. Complete 
the elevation by drawing a wire nail, Fig. 116, which is to be 
"soldered" in after the cone is formed. 

Profile of the Cone. — The profile of any cone of revolution is a 
true circle. This circle may be divided into two equal parts; 
therefore, it is necessary to draw but one-half of the profile as 
shown in Fig. 116. This profile is divided into equal parts and 
each division numbered. Extension lines are carried downward 
until they meet the base line of the elevation. 

Elements of a Surface. — Dotted lines are shown in Fig. 116 run- 
ning from each intersection of the base, to the apex. These repre- 
sent imaginary lines drawn upon the surface of the cone. If lines 
were drawn upon the surface of the cone, until the surface was 
completely covered, each one of these lines would become a part, 
or an element, of the surface. Any surface may be regarded as 
being made up of an infinite number of lines placed side by side, 
each line being an element of the surface. 



CONES OF REVOLUTION 



81 



The slant height Unes are also elements of the surface of the 
cone, and are the only elements shown in the elevation that repre- 
sent the true distance from the base to the apex, along the surface 
of the cone. 

The Arc of Stretchout. — With one point of the compass on the 



Tore shortened elements 
of aucfoce 

Slanf Height 

True length of element 



The arc of stretchout 

Elements of surface 

3 




Figs. 115-117. — Conical Flower Holder 



apex, and a radius equal to the slant height, an arc, Fig. 117, is 
drawn. Upon this arc, as many spaces are laid off as there would 
be in the whole profile. Fig. 116. Since this arc answers the same 
purpose as did the line of stretchout in parallel line drawing, it 



82 SHEET METAL DRAFTING 

can be called the arc of stretchout. The intersections on the arc 
of stretchout are numbered to correspond to the profile. Points 1 
are connected with the apex and j-inch edges are added for a tin 
lock. Notice that the locks are parallel to lines number 1 and do 
not connect with the apex until they are ''notched." The 
notch at the apex of any cone is made very long in order to bring 
the cone to a sharp point. The elements of the surface should 
be drawn, as shown in Fig. 117, and the length of these compared 
with the length of the corresponding foreshortened elements of 
Fig. 116. 

38. Related Mathematics on Conical Flower Holder. — Area of 
a Sector. — A sector is a part of a circle set off by two radii and an 
arc. Is Fig. 117 a sector? What is the length of its radius? 
What is the length of its arc? If the length of the arc is 
multiplied by one-half of the radius, the result will be the area 
of a sector. Thus the area of a sector whose arc measures 14" 
and whose radius is 7" would equal 14X3| = 49 sq. in. 

The length of the arc, Fig. 117, is equal to the circumference of 
the base of the cone (the profile). In addition, the radius of 
pattern. Fig. 117, is equal to the slant height of the cone. Fig. 116. 
Because of these facts, the formulae for the area of a sector and for 
the lateral area of a right cone are very much alike. 

Lateral Area of a Cone. — The lateral area of a right cone is equal 
to the circumference of the base times one-half the slant height. 

Suppose the base of a cone is 5" in diameter and the slant 

height is 12". To find the lateral area, the circumference, which 

would equal 5X3.1416=15.7080", must first be found. Then 

.1 <■ 1 J- A ^,, Slant Height 
usmg the lormula lor area, A = CX ^ — 

A = 15. 708 X V- 
^=94.258 sq. in. 

ProUem 19 A. — How many square inches of surface area 
(lateral) has a right cone whose base is 7" in diameter and whose 
slant height is Hi"? 

Problem 19 B. — The base of a cone has a circumference of 96" 
and a slant height of 102|". What is the area of its lateral 
surface in square inches? 

Problem 19C. — How much would 1000 flower holders, Fig. 115, 
weigh if made from No. 28 galvanized steel (.7812 lb. per square 
foot)? Allow 5 per cent of total weight for waste. 



CONES OF REVOLUTION 83 

Problem 20 
PITCH TOP COVER 

39. The Pitch Top Cover. — The cover of any receptacle should 
be made with a pitched top, such as considered in this problem, 
in order to obtain the necessary rigidity. 

The Elevation. — The elevation of the pitch top cover appears 
as shown in Fig. 119. This cover consists of a cone top, joined 
to a cylindrical rim by a "chnched" seam. The rim has a No. 
12 wire rolled into the bottom edge. A semicircular wired handle 
is drawn with a 2-inch radius by using the apex of the cone 
as a center. The distances C to B, and J to K, are straight lines 
connecting the semicircle and the slant height lines. The dis- 
tance from A to 5 is 1 inch. The handle is joined to the cover 
by 1| lb. rivets. 

The Profile. — A half-profile should be drawn, using extension 
lines to locate the view properly. The half-profile is divided into 
equal parts and each division numbered. The profile is equal in 
diameter to that of the rim inside of the wire. Extension hnes 
are carried upwards from each division of the half -profile, to -the 
base of the cone, and thence to the apex. 

Drawing the Pattern. — With a radius equal to the slant height 
of the cone, and any point as a center, the arc of stretchout is 
drawn. The spacing of the half-profile is transferred to the arc 
of stretchout, doubling the number of spaces in order to obtain the 
whole pattern. The divisions are numbered as shown in Fig. 120. 
A |-inch edge parallel to the arc of stretchout is added to allow 
for joining the rim. The locks are drawn parallel to lines 1 and 1, 
Fig. 120. These locks are notched as indicated. The rivet holes 
may be located on any two elements that are opposite each other, 
viz., on 4 and 4, 3 and 5, 2 and 6, etc., but in shop practice they are 
generally placed 90° from the lock seam. This would bring them 
on lines 4 and 4 as in Fig. 120. The distance from the center of 
the pattern to the center of the holes is found by measuring down- 
wards on the slant height. Fig. 120, from the apex of the cone 
to the center of the rivet as shown in the elevation. 

Pattern for the Rim. — The pattern for the rim, Fig. 121, is a 
straight piece of metal the length of which is equal to DXtt (diam- 
eter of profile X 3. 1416), and the width of which is equal to the 



84 



SHEET METAL DRAFTING 



F»G. 118 



§ Double edge 



Arc of 
stretch 00+ 




Figs. 118-122.— Pitch Top Cover. 



CONES OF REVOLUTION 85 

depth of the rim plus an allowance of I in, for the wire edge, and 
Ye in. for a single edge, A 1-inch lap for riveting must be added 
to the length. This lap must be so notched that it will not inter- 
fere with the wire, which is placed in position before the rim is 
formed into a cylinder. 

Pattern for the Handle. — Any straight Hne, Fig, 122, may be 
used as a line of stretchout. The profile of the handle, Fig, 122, is 
divided into equal parts and the divisions lettered A, B, C, etc. 
These divisions are transferred to the line of stretchout. Perpen- 
diculars to the line of stretchout are erected at points A, B, K, 
and M. Distances of | in, and I in, are set off on lines B and K, 
on each side of the line of stretchout. These intersections are con- 
nected by straight lines to form the body of the pattern. Lines 
A and M set off that part of the handle that laps and rivets to the 
cover, and should be notched as shown in Fig, 122. Rivet holes 
are located in the exact center of these laps. 

40. Related Mathematics on Pitch Top Cover. — Problem 20 A. 
— How much would 50 cone tops, shown in Fig. 118 (no rims or 
handles) , weigh if made from No, 26 galvanized steel ( . 9062 lb. 
per square foot)? Add 25 per cent for waste. 

Problem 20B. — What would be the weight of the cone top of a 
cover to fit over a 14" garbage pail? Allow |" clearance between 
pail and cover on all sides, making the diameter of the base of 
cone 15", and the slant height 9". Cover to be made of No. 26 
galvanized steel. 



86 SHEET METAL DRAFTING 

Problem 21 
VEGETABLE PARER 

41. The Vegetable Parer. — Figure 124 shows an elevation of a 
vegetable parer which is in the form of a right cone cut by a 
curved plane. 

The Elevation. — A "vertical" line which is to be used for the 
center line, or altitude of the cone, is drawn first. At right angles 
to the lower end of this line, the base of the cone is drawn. This 
base is to be f in. long and is to have f in. on each side of the center 
line of the cone. A distance of 17 in., which will locate the apex 
of the cone, is set off upon the center line. The apex and the ends 
of the base line are connected by the slant height lines. At an 
altitude of 4 in. a curve that cuts the cone, as shown in Fig. 124, 
is drawn in. This curve may be drawn to suit the ideas of the 
designer, and is in reahty the miter line. 

The Profile. — A whole profile, using extension lines to locate 
the view, is drawn. The profile is divided into twelve equal parts 
and each division numbered. Extension lines are carried from 
each division of the profile upwards to the base of the cone, and 
thence to the apex. Each one of these extension lines intersects 
the miter line at some point. Horizontal lines from each of the 
intersections of the miter line, Fig. 124, are drawn over to the slant 
height. 

The Pattern. — The arc of the stretchout, Fig. 125, using the 
apex as a center and a radius equal to the slant height of elevation, 
is next drawn. The spacing of the profile is transferred to the arc 
of stretchout and the divisions numbered to correspond. From 
each of these points a measuring line of the stretchout is 
drawn to the apex. Starting at point 1 of the profile, the 
extension line should be followed up to the miter line, then 
horizontally to the slant height? With a radius equal to the 
distance from this point to the apex, and the apex as a center, a 
curved extension line should be drawn over into the stretchout 
until it intersects both lines that bear the number 1. (Is this 
procedure similar to that of parallel line drawing? Wherein does 
it differ?) In like manner, each intersection on the stretchout 
may be traced out. A curved line passing through these points 
will give the miter cut. A j-inch lap is added to one side of the pat- 



CONES OF REVOLUTION 



87 




Figs. 123-125.— Vegetable Parer. 



88 SHEET METAL DRAFTING 

tern as shown in Fig. 125. A |-inch hem should be added below 
the arc of stretchout. The slot for paring purposes is laid out 
on hne 7. Starting I in. from the smail end, a distance of two 
inches is laid off on hne 7. The slot being -^ in. wide will require 
■^ in. on each side of line 7. One edge of the slot is shghtly 
raised, after the parer is formed, and then filed to a cutting edge. 



CONES OF REVOLUTION 89 

Problem 22 
CONICAL ROOF FLANGE 

42. The Conical Roof Flange. — Whenever a smoke pipe is to 
pass through a roof, it is necessary that a hole much larger than the 
pipe be cut in the roof in order to lessen the fire risk. In order to 
render this construction water-tight, a conical roof flange, as 
shown in Fig. 126, must be used. 

The Elevation. — First, the roof line is drawn at the angle 
demanded by the job specifications (ia this drawing 30°). The 
roof fine immediately becomes the miter line. Next, a "vertical" 
center Hne, fine 4 in Fig. 127, is drawn in. Upon each side of this 
center line a distance equal to one-half the diameter of the smoke 
pipe is set off. A short horizontal line is put in to represent the 
joint between the pipe and the flange. One-half the diameter of 
the hole in the roof is set off on each side of the vertical center line. 
This will locate the low point (point 7) of the miter line. From this 
point the base of the cone should be drawn at right angles to the 
center line. The slant height lines may now be drawn by con- 
necting the ends of the base and the ends of the short horizontal 
line of the junction between the flange and pipe. These lines must 
be prolonged until they meet at the apex. 

The Profile. — A half-profile. Fig. 127, is drawn and divided into 
equal spaces. The divisions are numbered and an extension car- 
ried upwards from each division as far as the base of the cone. 
From each intersection thus obtained, extension lines are drawn 
to the apex of the elevation. Where these extension fines cross 
the miter line, numbers that correspond to the numbering of the 
profile are placed. Horizontal extension lines from each inter- 
section of the miter line are drawn over to the right-hand slant 
height fine. 

The Pattern. — With the apex of the elevation as a center, and a 
radius equal to the slant height of the full cone, the arc of stretch- 
out is drawn. The spacing of the profile is transferred to this 
line, and the number of spaces doubled to provide for a whole 
pattern. The divisions are numbered to correspond. The 
measuring lines of the stretchout are drawn from each division of 
the arc of stretchout to the center point (apex). 

Starting from point 1 of the profile, the extension lines are 



90 



SHEET METAL DRAFTING 



Fi6. 126 



^Double 




y Arc of 
s<Te+chou+ 



Figs. 126-128— Conical Roof Flange. 



CONES OF REVOLUTION 91 

traced to the base of the cone, then to the miter hne. With a 
radius equal to the distance from this point to the apex, and with 
the apex as a center, a curved extension hne intersecting hnes 1 and 
1 of the stretchout is drawn. From point 2 of the profile, the 
extension line is traced to the base of the cone, then to the miter 
line, and thence horizontally to the slant height line. With a 
radius equal to the distance from this point to the apex, and with 
the apex as a center, a curved extension line intersecting lines 2 
and 2 of the stretchout is drawn. In like manner, the remaining 
intersections of the stretchout may be traced. A curved line 
passing through these points will give the miter cut of the roof 
flange at the roof line. The upper miter line, being parallel to 
the base, is developed like an ordinary right cone. With a radius 
equal to the slant height, a curved extension line passing through 
the stretchout is first drawn. The necessary allowances for 
locks parallel to lines 1 of the stretchout are added. One- 
half inch double edges to the upper and lower miter cuts of 
the pattern are also added to allow for joining to the pipe and 
apron. 

43. Related Mathematics on Conical Roof Flange. — Area 
of Frustum. — If a right cone is cut by a plane parallel to that of the 
base, the top section will stiU be a right cone although of small 
dimensions, and the lower part will be a frustum of a cone. The 
profiles of both bases, or ends, will be circles. The smaller circle 
is generally called the upper base, and the larger circle the lower 
base of the frustum. The lateral area of a frustum of a cone is 
found by adding together the circumferences of the upper and 
lower bases, dividing the sum by 2, and then multiplying by the 
slant height. This is often expressed as a formula for area of a 
frustum : 

in which 

B = Circumference of lower base 
6 = Circumference of upper base 
Hs = Slant height of frustum 

A frustum whose lower base has a circumference of 40" and 



92 SHEET METAL DRAFTING 

whose upper base has a circumference of 22" will have a surface 

area of 

40+22 

o Xl6 = 496sq. in. 

if the slant height of the frustum is 16". 

Also, a frustum having an upper base diameter of 8", a lower 
base diameter of 12", and a slaijt height of 10" will have a 
surface area of: 

Circumference of upper base = 8" X 3 . 1416 = 25 . 133" 

Circumference of lower base = 12" X3 . 1416 = 37 . 669" 

Formula, A =- — — — X Hb 

2 

^ , ^., ^. , 37.699"+25.133"^^,„ 
Substitutmg, A = XIO 

A = 314. 16 sq. in. 

Problem 22 A. — The roof flange, Fig. 126, would be treated by 
any estimator as a frustum of a right cone, although in reality 
its surface area is less. The estimator would take the diameters 
of the upper and lower bases from the elevation. Fig. 127, calHng 
the upper base 2" and the lower base 6" in diameter. How much 
would the conical part of this roof flange weigh if it were made 
from No. 26 galvanized iron? 



CONES OF REVOLUTION 93 

Problem 23 
APRON FOR A CONICAL ROOF FLANGE 

44. The Apron for a Conical Roof Flange. — When any solid is 
cut by a plane that is inclined to the plane of the base, the shape 
or section thus formed is not the same as the profile of the base. 

The Elevation. — The elevation of the roof flange used in Prob- 
lem 22 can be reproduced. 

The Pattern. — Any straight line, Fig. 130, may be drawn and 
the exact spacing of the miter line set off upon it. Perpendic- 
ular measuring lines, Fig. 130, are erected through each point 
and are numbered to correspond to the miter line. 

Returning to the elevation, Fig. 129, an extension Hne is 
dropped from point 1, down to the horizontal center line of the 
half-profile. From point 2 of the miter line, the horizontal exten- 
sion line is followed over to the slant height. From this point, a 
perpendicular to the horizontal center hne of the profile is dropped, 
and with one point of the compass on the center of the profile, this 
line is extended by an arc until it strikes a radial line from point 2 
of the profile at the point B. 

The perpendicular distance from point B to the horizontal 
center hne should be measured and placed on each side of the line 
of stretchout. Fig. 130, on measuring hne number 2. In hke 
manner, points C, D, E, and F are located and their distances 
placed on measuring fines 3, 4, 5, and 6 respectively. A curve 
traced through the points thus obtained will give the shape of the 
hole in the apron as well as that of the hole to be cut in the roof. 

A rectangle representing the shape of the apron should be 
drawn, allowing a space of at least 6 in. "up the roof," and at 
least 3 in. on the other sides. A hem should be added to three 
sides to turn or direct the flow of any roof water that might leak 
in. A -j^-inch single edge should be allowed around the inside of 
the hole, in order to double seam the body to the apron. 



94 



SHEET METAL DRAFTING 



Fig. IZ9 




Fig. 130 ^^^^ 

Figs. 129-130.— Apron for Conical Roof Flange. 



CHAPTER VI 
INTERSECTING RECTANGULAR PRISMS 



Prob 
No. 



JOB 



Draw I n g 

OBJE-CTIVE 



Mathematical 
Objective 



24 










SquarePipe Elbow 



25 




^rawing ^e 
Af/'fer jL//7es. 



^rea o/ Para/Ze/- 



5<5UARE Pipe Offset 



25 




0/7 ecZ^e.. 






QiAGONAU 

Offset in 
Square Pipe 



27 




Meffiods of 
Co/f sfruch'on. 



Area o/^ Cy/Zn^r/ca/ 
of f?evo/of/'<?n 



Curved Elbow 
IN Square Pipe 



Objectives of Problems on Intersecting Rectangular Prisms. 

95 



96 SHEET METAL DRAFTING 

Problem 24 
THREE-PIECE RECTANGULAR ELBOW 

45. The Three-piece Rectangular Elbow. — This problem deals 
with a three-piece, 90° elbow, Fig. 131, having a throat radius of 
4| in. Looldng at the elevation of this elbow, Fig. 132, one would 
be unable to tell whether the fitting was round or rectangular 
piping. The profile shows that the elevation is of a rectangular 
pipe elbow. 

The Elevation. — First, an angle of 90°, one side of which is to 
be used as the base line of the elbow, is drawn. From the vertex of 
the angle, distances of 4^ in. and 2| in., as shown in Fig. 132, are 
set off. The arc of the throat and the arc of the back are drawn, 
using the vertex of the angle as the center of the elbow. The arc 
of the back is divided into four equal parts. Miter lines are 
drawn through the first and third divisions of the arc, above the 
base Hne. The elevation is completed by drawing straight lines 
tangent to the arcs. The detailed description of an elbow eleva- 
tion is given in Problem 10, Chapter III. 

The Profile. — A profile, Fig. 133, is drawn, using extension Hues 
to locate the view properly. Each corner of the profile is num- 
bered. It should be remembered that the seam always occurs at 
number 1 in the profile. In this case the seam comes at one 
corner of the elbow. Many prefer to have the seam at the center 
of one of the sides, or faces, of the elbow. 

The Patterns. — Three Unes of stretchout, one at right angles 
to each piece. of the elbow, must be drawn. The spacing of the 
profile is transferred to each line of stretchout and is numbered 
to correspond. Measuring lines are drawn through each point 
in the lines of stretchout. Starting at point 1 of the profile an 
extension hne should be traced upwards to the miter lines, and 
from there an extension line to lines 1 of the stretchouts should be 
drawn. Notice that two stretchouts are served by each inter- 
section of the miter line. Extension lines from the elevation into 
any stretchout must always be drawn at right angles to the sides 
of the pipe. In like manner intersections of the stretchouts can 
be located and the patterns completed by drawing straight lines 
between these points as shown in Figs. 134, 135, and 136. Three- 
sixteenths inch single edges and |-inch double edges, as shown, 
join the pieces of the elbow by double seaming. 



INTERSECTING RECTANGULAR PRISMS 



97 



46. Related Mathematics on Elbows. — Solids of Revolution. — 
All elbows may be treated mathematically as solids of revolution. 



^'Double edge 



|[ Double edge 




ri 



^!;'2?'f Pattern OF First Piece OoM^ ^dQ^r^ 



Profile 

Fl<3. 

133 



edge 



Figs. 131-136. — Three-piece Rectangular Elbow. 



Any surface moving about a fixed point will generate a solid of 
revolution. Suppose a piece of round rod is formed in the rolls 
to a true circular profile. A soHd of revolution would be created 



98 SHEET METAL DRAFTING 

because a ring slipped over the rod and caused to move around it 
to the right or to the left would always be the same distance from 
the center of the profile to which the rod was formed. A piece of 
bar iron formed to a circular profile would also be a solid of revo- 
lution. This solid could be regarded as being generated by a 
rectangle revolving about a center point. Figure 132 is drawn 
around two arcs whose center is the center of the elbow. It can 
also be seen that the more pieces "put in" the elbow, the nearer 
the straight lines come to the arcs about which they are drawn. 
If the elbow were made of a very great number of pieces, these 
would become so much like arcs that they could hardly be dis- 
tinguished from them. This or any other elbow can be treated as 
a solid of revolution. 

Solids of Revolution Have Three Diameters. — Every solid of 
revolution may be considered as having three diameters. The 
radius of the throat of Fig. 132 is 4^". If four of these elbows were 
joined so as to make a complete ring, it would have a diameter 
of 4|"X2, or 9". This would be the inside diameter of the solid 
of revolution. The radius of the back, Fig. 132, is 4i"4-2|", or 7". 
The corresponding diameter for the whole ring would be 7" X 2 or 
14". This would be the outside diameter of the solid of revolu- 
tion. The third diameter is twice the center line radius of the 
elbow. In Fig. 132 the center line radius is 5|", and for the whole 
ring this would be 5|"X2 = 11|". This is called the neutral 
diameter of the soUd of revolution, because when any rod or bar is 
formed into a circular profile the metal near this line stands still, 
that outside of the line stretches, and that inside of the line shrinks 
a like amount. This can be proved by drawing straight lines on a 
pencil eraser and bending the eraser, at the same time noting the 
distances between the lines. 

Rule for Surface Area. — The surface area of a solid of revolution 
is equal to the circumference of its right section (profile) multiplied 
by the length of its neutral zone (diameter of the neutral X3 . 1416). 

Sample Problem. — What is the surface area of the elbow shown 
in Fig. 132 ? 

Outside diameter of ring = 14" 
. Inside " " " - 9". 

Neutral " '' =1H". 

Length of neutral zone (ll|"X7r) =36. 128". 

Perimeter of right section (length of line of stretchout) = 20". 



INTERSECTING RECTANGULAR PRISMS 99 

Surface area of entire ring 36 . 128" X 20" = 722 . 56 sq. in. 

Surface area of elbow (90° or I of entire ring) 722 . 56 sq. in. -i-4 
= 180.64 sq. in. 

Ans. 180.64 sq. in. 

Problem 2Ii.A. — A 90° elbow to fit a rectangular pipe 7"X12" 
has a 10|" throat radius. What is its surface area? 

Problem 24B. — A 60° elbow (^ of the solid of revolution) to 
fit a rectangular pipe 30"X61" has a throat radius of 17". What 
is its surface area? 



100 SHEET METAL DRAFTING 

Problem 25 
RECTANGULAR PIPE OFFSET 

47. The Rectangular Pipe Offset. — As was pointed out in the 
preceding problem, the elevation of a fitting for rectangular piping 
presents much the same appearance as an elevation for a round 
pipe fitting. For this reason the description given below will 
answer equally as well for an offset in round piping, the only 
difference being the shape of the profile. 

The Profile. — A profile, as shown in Fig. 138, using the dimen- 
sions given, should be drawn. Each vertex (corner) of the profile 
should be numbered. Extension lines are carried upwards from 
points 2 and 3. 

The Elevatio7i. — A base Une AH in Fig. 137 should be drawn 
equal in length to line 2-3 of the profile. A perpendicular 3 
inches high should be erected at point A. The upper point of this 
fine should be lettered B. The line BC is drawn according to 
the cUmensions given in Fig. 137. The 3-inch dimension gives 
the fitting its name, it being the amount that the third piece sets 
off to one side of the first piece of the elbow. The Une CD is 
drawn at right angles to the base fine. This completes the out- 
hne of one side of the elevation. The miter fine BG must next be 
drawn. Every miter fine of an elbow bisects the angle formed by 
the adjacent sides of the pipe. Therefore, in order to get the miter 
line BG the angle ABC must be bisected. 

The procedure for bisecting an angle is as follows: With B 
as a center and any radius, set off equal distances on each side 
of the point B, on lines AB and BC. Letter these points P and 
R. With P and R as centers and any radius greater than RB, 
draw intersecting arcs (to the right of the figure). Letter this 
intersection J. 

The straight fine BJ will bisect the angle and can be used as the 
first miter fine of the fitting. A perpendicular is erected at the 
point H until it cuts the miter line at the point G, Fig. 137. The 
next fine to be drawn is FG. 

In order to draw the line FG the angle HGB must be copied. 
This is done as follows: With (? as a center and any radius, draw 
an arc cutting fine HG at the point K and also cutting the miter 
line at the point M. With ikf as a center and a radius equal to 



INTERSECTING RECTANGULAR PRISMS 



101 



MK, set off a distance equal to MK on the other side of the miter 
line. Letter this point N. The straight hne GN will form the 
angle NGB, which will exactly equal the angle HGB. Make the 
line GF equal in length to hne BC. Draw the hne CF, which is 
the second miter line of the elevation. 



■g Double edg* 



Cut on these 
ines 




Fig. 141 



Figs. 137-141.— Rectangular Pipe Offset. 



The elevation is completed by drawing lines DE and EF. 

The Pattern. — There are two methods of construction in general 
use in shop practice. One method calls for each piece to be devel- 
oped separately, Fig. 139, the other calls for the body of the fitting 
to be made as shown in Fig. 140, the ends being "double seamed 
in." Figure 139 shows the patterns for the offset laid out in 



102 SHEET METAL DRAFTING 

such a manner that no stock will be wasted. The measurements 
for the several pieces are taken from the elevation and are plainly- 
marked. The pattern for the body piece, Fig. 140, is obtained by 
adding 3-inch double edges to the front elevation, and notching 
for a If-inch cleat as shown. The stretchout for the end piece 
is determined from the front elevation, and the width from the 
profile. Single edges I in. wide are added to each side. The 
top is notched for a l|-inch cleat. 

48. Related Mathematics on Rectangular Pipe Offset. — 
Problem 25 A. — How much would the stock cost for four rec- 
tangular pipe offsets, Fig. 137, made from No. 24 galvanized steel 
(1.156 lb. per square foot) if the steel cost $8.75 per 100 lb.? 

Problem 25B. — How much more would the stock cost for the 
patterns shown in Fig. 140 than those shown in Fig, 139? Stock 
cut from body pattern corners would be regarded as waste. 



INTERSECTING RECTANGULAR PRISMS 103 

Problem 26 
DIAGONAL OFFSET 

49. The Diagonal Offset. — This type of fitting is used in venti- 
lating and heating ducts. It is also frequently encountered in run- 
ning rectangular copper conductor pipes. The elevation shows a 
section of a lintel cornice such as is frequently seen above the first 
floor windows of a building. A conductor pipe running down an 
inside corner of a building having a lintel cornice would have to 
be offset diagonally in order to clear the obstruction. 

The Plan. — Figure 142 shows the outline of a conductor pipe 
(fines AB, BC, and CD). The entire elevation is not shown 
because it plays no part in the development. The plan, Fig. 143, 
is drawn making the angle equal to 45°. If the required angle is 
other than 45°, it presents an entirely different problem and can- 
not be drawn by this method. The profiles are numbered as 
shown. Profile 1, 2, 3, and 4 is that of the lower part of the fitting 
and profile 5, 6, 7, 8, that of the upper part. While both are the 
same size their numbering must be different. 

Diagonal Elevation. — A base fine for the diagonal elevation 
must be drawn parallel to the line 4-8 of the plan. Extension 
lines are carried from each point in both profiles at right angles to 
this base line. Extension lines from points 1 and 3 will locate 
points A and E on the base line. A perpendicular is erected at 
point A and the distance AB set off equal to AB of the elevation, 
Fig. 142. The extension line from point 5, of Fig. 143, will locate 
points C and D in Fig. 144, heights being taken from Fig. 142. 
Drawing the line BC wiU complete the outline of that part of the 
fitting that rests directly upon the lintel cornice. A perpendicular 
is now erected at point E of Fig. 144. With i? as a center and a 
radius equal to AE, an arc is drawn as shown in Fig. 144. Any 
other point F on the line CB is selected and another arc of the 
same radius drawn. The fine HG must be drawn tangent to both 
arcs. The point G occurs at the intersection of this tangent and 
the extension fine from point 3 of Fig. 143. Line GH should be 
equal in length to BC. At the point D a line is, drawn at right 
angles to hne CD. This line wUl intersect the extension line 
from point 7 of Fig. 143. The intersection should be lettered K. 
A straight line KH will complete the outfine of the diagonal* 



104 



SHEET METAL DRAFTING 



elevation. The miter lines BG and CH should be drawn in. 
The solid and dotted Hnes representing the other edges of the pipe 
are located by extension hnes running from points 2, 4, Q, and 8 of 




Figs. 142-147— Diagonal Offset. 

Fig. 143. The patterns are drawn in the manner described in 
Problem 25. 

Locks are purposely left off of the patterns in order to avoid 



INTERSECTING RECTANGULAR PRISMS 105 

a confusion of lines. It is recommended to the student that he 
make paper or metal models of this problem in order to understand 
thoroughly the basic principles involved. 

50. Related Mathematics on Diagonal Offsets. — Problem 26 A. 
— Sixteen-ounce cold-rolled copper costs 25j cents per pound. 
How much would twenty-four diagonal offsets, Fig. 144, cost? 
In finding the area of this fitting multiply the distance (perime- 
ter) around the profile by the combined length of lines AB, BC, 
and CD of Fig. 142. Sixteen-ounce copper weighs 16 ounces to 
the square foot. Allow 5 per cent for locks. 



106 SHEET METAL DRAFTING 

Problem 27 
CURVED ELBOW IN RECTANGULAR PIPE 

51. The Curved Elbow in Rectangular Pipe. — This type of 
fitting is extensively used in ducts for heating and ventilating 
systems because it offers a minimum of friction to the moving 
air. The elbow discussed here has an angle of 90°. 

The Profile. — A plan or profile, Fig. 148, is drawn according to 
the dimensions given. The corners are lettered, and extension 
lines are carried upwards from points A and D to locate the eleva- 
tion properly. 

The Elevation. — After a base line is drawn, points 1 and 12 are 
located, and a distance of 3f in. (to scale) set off to tbe left of point 
1 to serve as the center point of the elbow. The limits of the 
elbow are defined by erecting a perpendicular at this point. The 
arcs of the throat and the back are drawn. These arcs are divided 
into equal spaces and are numbered as shown. 

The Pattern. — The pattern for the body, Fig. 150, is a copy of 
the elevation. To this is added a |-inch single edge on the throat 
and back. A IJ-inch edge for "shipping" the pipe is added as 
shown. Figure 151 shows the pattern of the throat piece which 
is a rectangle whose width is equal to line AB of plan, and whose 
length is equal to the stretchout of the spacing of the arc of the 
throat. Fig. 149. To this rectangle must be added 1| in, for 
"shipping" and to each long side a j^-inch edge for a hammer 
lock. Figure 152 differs from Fig. 151 only with regard to its 
length, which is taken from the stretchout of the arc of the back. 

The Hammer Lock. — The hammer lock is so called because it 
can be made up on the job, the only tool required being a hammer. 
Straight strips of metal are formed in the brake, to act as the sides 
of the fitting as shown in Fig. 153. The |-inch single edges of the 
body are worked up to a right angle and slipped into the slot of the 
hammer lock. The protruding edge is then closed down with a 
hammer as shown in Fig. 154. This gives the job an appearance 
of being double seamed, and requires much less time and effort 
than the double seamed job. 

52. Related Mathematics on Curved Elbows. — Problem 27 A. 
— How much would three curved elbows, Fig. 149, weigh, made 
from No. 28 galvanized iron (.7812 lb. per square foot), allowing 
20 per cent for waste? 



INTERSECTING RECTANGULAR PRISMS 107 



Hammer lock- 



M 


+ 








f 




•J 


III 




4. 






« 




fc, 


4- 


♦ 




4- 


n 











« 


ti 




c 




' 


-1 


_ 



FiQ.151 



« — ?-»] 
Fi<5.'l5Z 



hl*h4*l 



Side 



"Hommar lock 
(open) 



Fig. 153 b 



ody- 



Side 



Fig. 154 

Body^ 



-1 Hammer lock 
'(closed) 




•■el ■ ■* 

Ship 




e ed9« 



Figs. 148-154.— Curved Elbow in Rectangular Pipe. 



CHAPTER VII 
PLANNING FOR QUANTITY PRODUCTION 



Prob. 


Job 


DR AWI NG 


Mathematical 


No. 


Objective 


OBJECTIVE 




f 


c: 


n — rH 


1 


/>e/^a////7^ opera f/om 
forroi/f//rf mroi/_^/) 
ff?e 5/?op as p/ecetvork 
Sfand&rd comfrvcf/of?. 


£sf//77af//7^s cosfs. 


28 


k 


© 




. 


) 












CAN. 







Objective of Problem on Quantity Production 
109 



110 SHEET METAL DRAFTING 

Problem 28 
ASH BARREL 

53. Planning for Quantity Production of an Ash Barrel.— 

In planning for quantity production, a draftsman must consider 
every item that has to do with the manufacturing processes to be 
carried on in the shop. In order to do this intelligently he should 
make a list of these items similar to the one given below: 

1. Dimensions of barrel 

2. Type of hoop to be used 

3. Blank for body: 

a. Weight of material 

h. Height after deducting hoops 

c. Riveted or locked seams 

4. Weight of bottom 

5. Type of slat to be used 

6. Number of slats required 

7. Sizes of rivets required: 

a. For attaching slats 

6. For attaching upper rim 

c. For attaching lower rim and bottom 

8. Pattern for body: 

a. Over-all dimensions 
6. Allowance for lock 

c. Locating rivet holes for slats 

d. Locating rivet holes for upper hoop 

e. Locating rivet holes for lower hoop 

9. Pattern for upper hoop and lower hoop: 

a. Length of blank required 
h. Method of joining ends 
c. Spacing rivet holes 

10. Pattern for bottom: 

a. Allowance for flanging 

11. Pattern for slat: 

a. Miter cut 

h. Size and location of rivet holes 

c. Size of wooden core 

12. Order in which parts are assembled 

Dimensions of Barrel. — The dimensions of the barrel would vary 
according to whether the job was standard, or special size. Dif- 
ferent manufacturers have estabhshed their own standards. The 
sizes given below are common to all, and are best adapted to ordi- 
nary needs. A barrel 18 in. in diameter by 26 in. high will be 
treated in this discussion. 



PLANNING FOR QUANTITY PRODUCTION 111 



( 






S+raighi Edge 



5ec + ion Thirouqh Barret 
Fig. 157 




4 Ti'nners Civet 




Fig. J56 



e^Tinnar'a giygf 



Three Bib Slat 
Fig. 158 



Figs. 155-158. — Galvanized Ash Barrel. 



112 SHEET METAL DRAFTING 

Standard Dimensions for Galvanized Ash Barrels 



Diameter 
in Inches 


Height 
in Inches 


Approximate Capacity 


24 

20 

18 

17i 

14 


36 • 

26 
26 
26 
26 


72 gallons 

34 

28 

24i " 

16 



Type of Hoop. — There are several types of hoops that can be 
obtained from the jobber. Figure 156 gives a full size cross 
section of the one generally used. 

Blank for Body. — The body of the ash barrel is made from No. 
24 galvanized steel. The top and bottom hoops fit into the barrel 
one-half of their entire width, Fig. 157; therefore, the body blank 
must be | in.X2 or If in. less than the total height of the barrel. 
This would make the blank 24| in. wide, but in order to use the 
sheet metal as it comes from the mill the total height would be 
reduced to 25f in., Fig. 157, and stock size sheets 24 in. wide used 
to make the body. The riveted seam is somewhat stronger, but 
since the lock seam can be placed under a slat and protected, it is 
generally used because it can be made more cheaply. 

Weight of Bottom. — The bottom of the barrel should be at least 
four gages heavier than the sides. 

Type of Slat. — Figure 158 shows two types of slats in common 
use. The three-rib slat is made of No. 24 galvanized steel by 
means of special machinery. The single-rib slat may be made on 
an ordinary cornice brake and used with or without the wooden 
core. 

Sizes of Rivets. — In riveting the slats to the barrel, the rivets 
must pass through two thicknesses of No. 24 gage. This will re- 
quire a 2| lb. rivet. The rivets for the upper hoop must pass 
through one thickness of No. 24 gage iron and ^ in. of steel in the 
hoop. This will require a 6 lb. rivet. The rivets for the bottom 
hoop must pass through one thickness of No. 24 gage (the body), 
one thickness of No. 20 gage (the bottom), and ^ in. of steel in 
the hoop. This will require an 8 lb. rivet. 

Drawing the Section. — A section. Fig. 157, showing the hoops, 
body, and bottom in their proper positions, should be drawn. 



PLANNING FOR QUANTITY PRODUCTION 113 

The Profile. — The profile of the body should be drawn. The 
profile of the hoops would be larger than that of the body, but since 
the pattern for the body and the location of the slats are to be 
obtained, the profile of the body must be dealt with. This profile 
is divided into eight equal parts. Fig. 155. Using any two divi- 
sions of the profile as centers, the two slats are drawn in their 
proper positions. The profile. Fig. 155, shows the three-rib slat, 
but a single-rib slat may be drawn in. A straightedge laid 
across the slats, in the manner shown in Fig. 155, should clear 
the body of the barrel; otherwise, the slats will not protect the 
body when in contact with the edge of the ash cart. If the 
straightedge touches the profile, the slats must be made larger 
or spaced more closely together. 

Pattern for the Body. — A line of stretchout, Fig. 159, should be 
drawn and the spacing of the profile transferred to it, with num- 
bers to correspond. Measuring lines are drawn through each 
division of the stretchout. One-half inch locks are set off on each 
end of the pattern. The top and bottom lines of the pattern are 
drawn 24 in. apart. A distance of j^ in. should be measured in from 
the top and the bottom lines. These fines will serve as center lines 
for the rivet holes of the hoops and slats. From Fig. 158 the dis- 
tance from center to center of the ^-inch edge of the single-rib slat 
is found to be If in. This will also be the distance between centers 
for rivet holes in the body. One-half of this distance, y^ in., is 
placed on each side of every measuring line in the stretchout. It 
should be indicated on the drawing that these are to be punched 
for 2| lb. rivets. Another line running horizontally through the 
center of the pattern should bear the same spacing for riveting the 
center of each slat. The rivet holes for the upper and lower hoops 
are located midway between measuring lines 1 and 8, 7 and 6, 5 
and 4, 3 and 2 as shown in Fig. 159. This allows for four rivets. 
Therefore, the distance between centers for each pair of rivets will 
be one-fourth of the circumference. In order to avoid riveting 
through the seam, the first hole is spaced yg of the circumference, 
and the last hole j^ of the circumference away from the circum- 
ference line No. 1 of the stretchout. The rivet holes for the drop 
handles should be placed at f of the height of the barrel as shown. 
Fig. 157. 

Upper and Lower Hoops. — The upper hoop must be fitted inside 
of the body. In order for the hoop to go inside, some allowance 



114 



SHEET METAL DRAFTING 



j^f»_3of cli"CUin.-»+i — 1- of cWeumfei: — » | « ^ °^ circumfec— .^ io-f circomfar -^J^ cin f* 




Diam. yTH-locKif 

F»<s. 153 



Pattern for Upp^r Hoop 



-^ — ;j of circuniferei\ce4-8 o^ ^"'' 



Tk 



Vis 



4— ?- 



Fig. ISO. 



Pattern for Lower Hoop 




1 






J 





Fig. 161 




Body of borrel- 
Fig. 163 



Figs. 159-163. — Drawings for Patterns of Ash Barrel. 



PLANNING FOR QUANTITY PRODUCTION 115 

must be made for the thickness of the metal. Figure 156 shows a 
''dot and dash" line. This line is called the neutral axis, and takes 
its name from the fact that the metal at this point remains station- 
ary while that on either side stretches or shrinks as the hoop is 
formed up. It will also be noticed that this line is not in the center 
as it would be if the cross-section were rectangular in shape. 
According to Fig. 156, this line passes the ''square corner," against 
which the top of the body rests, at a distance of ^"— ^", or ys". 
The rule is to double this quantity and add one thickness of the 
metal body. Following this rule would give ys"-\-y6"-\- .025"= . 
.150" or -^" (nearly). This should be subtracted from the 
diameter of the body (18" — .15" = 17. 85") and the remainder 
multiplied by tt, in order to get the length of the blank for the 
upper hoop. Fig. 160. This would give 17.85" X3.1416 = 56.077", 
for the length of Fig. 160, The pattern for the lower hoop. 
Fig. 161, must be shorter than that for the upper hoop, because 
the lower hoop must go inside of the bottom of the barrel as well 
as the body. Fig. 157. Using the rule given above: 



-p^. _ /l thickness , 1 thickness allowance \ 

,„,', I body bottom for hoop L, oi^-.,> i xi. e 

'^ \ .025 + .037 + .125 ^ y^X 3.1416 = length of 

blank for lower hoop (Fig. 161) 
55.95" 

Since the ends of the hoops are to be butt-welded no allowance 
need be made for joining. There are to be four rivets in each hoop; 
therefore, the distance between the rivet holes on centers would be 
equal to one-fourth of the circumference. Placing half of this 
space, or ^ of the circumference, at each end would avoid a rivet 
hole through the weld. The spacing of the rivet holes in Fig. 160 
will not be the same as that in Fig. 161, because of the differ- 
ence in length of the blanks. 

Pattern for the Bottom. — The bottom of the barrel has a |-inch 
flange turned for riveting to the body. This flange can be worked 
up by hand, but it is generally pressed in a machine. Figure 162 
shows a section of the bottom, and the pattern with allowance 
for flanging. If | in. were added to each side of the diameter of 
the finished bottom, the machine would turn a flange deeper 
than I in. The rule for finding the correct diameter of the pat- 



116 SHEET METAL DRAFTING 

tern is: Find the total surface area of the finished piece and convert 
this area into disc inches. Applying this rule, 

Diameter^ X . 7854 = (18'02X .7854 = 

area of bottom = 254 . 47 sq. in. 
Circumference X Height = 18" X tt X 7/8 = 

area of flange = 49.48 sq. in. 
Total or combined area =303.95 sq. in. 



VArea ^ . 7854 = V303 . 95 -r- . 7854 = V387 = 19 . 67", 

Diam. of pattern. 

The nearest fraction to . 67 in. to be used would be |-| in. ; 
therefore, the diameter of the pattern for the bottom, Fig. 162, 
would be 19|~| in. 

Pattern for Slat. — The top and bottom of the slat are "cut 
back" on an angle of 60° as shown in Fig. 163. An elevation 
showing the miter cut at a 60° angle should be drawn. Extension 
lines are carried from the profile to the miter line. A line of 
stretchout is drawn and upon it the spacings of the profile are 
set off. The measuring lines are drawn in. The intersections 
from each point in the profile are traced to the miter line, and 
thence to the corresponding line of the stretchout. These inter- 
sections are connected by straight lines to obtain the miter cut. 
If the wooden core is to be used, some means for closing the end 
must be provided to prevent the core from slipping out. If the 
proper machine is available, an end may be "pressed on" the 
metal slat. Another method is to provide laps as shown by the 
dotted fines on the pattern. These laps may be fastened by one 
rivet. Holes for riveting the slats to the barrel must be laid out 
to correspond exactly to the spacing of holes in the body pattern, 
Fig. 159. 

Assembling the Barrel. — The body blanks are cut from sheets of 
No. 24 galvanized steel 24 in. wide by 120 in. long. Rivet hole 
centers are transferred from the master pattern, and holes are 
punched in each blank. Locks are then turned in the stovepipe 
folder, after which the body blanks are formed in the rolls and 
grooved in the grooving machine. The slats are riveted to the body. 
The upper hoop is then riveted on. The bottom and the lower 
hoop are then placed in the barrel and riveted in place. The drop 



PLANNING FOR QUANTITY PRODUCTION 117 

handles are attached to the barrel and it is then ready for final 
inspection. 

54. Related Mathematics on Ash Barrel. — In order to esti- 
mate the cost of the article to be made by quantity production 
methods, each item must be considered separately. 

Sample Prohlem. — Figure the cost of stock entering into the 
manufacture of 500 ash barrels such as shown in Fig. 157. 

Item 1. Cost of 500 Body Blanks. Fig. 159. 

Size of sheets 24" X 120", No. 24 gage 

Area of sheet 20 sq. ft. 

Number of sheets required — 250 (2 bodies from each sheet) 

Total area of 500 body blanks 250X20 = 5000 sq. ft. 

Weight of No. 24 galv. steel per sq. ft. = 1 . 156 lb. 

Total weight of bodies, 1 . 156 X5000 = 5780 lb. 

Cost of 500 bodies at 8.5?; per lb. =$491 .30 

Item 2. Cost of 500 Bottom Blanks. 

Size of blank = 19||" diameter 

Size of sheet required, 24" X 120' , No. 20 gage 

Number of blanks from each sheet, 6 

Number of sheets required, 500-^6 = 84 

Total area of each sheet = 20 sq. ft. 

Totalareaof 84 sheets (84X20) =1680 

Weight of No. 20 galv. steel per sq. ft. = 1 . 656 lb. 

Total weight of metal required (1680X1.656) =2782 lb. 

Cost of 500 bottoms at 8 . 5f5 per lb. = $236 . 47 

Item 3. Cost of 4000 Single Rib Slats. 

Size of blank, 24"X2i", No. 24 gage 

Size of sheet required, 24" X 120" 

Number of blanks from each sheet, 41 

Number of sheets required, 4000-^41 =98 sheets 

Weight per sheet, 23. 12 lb. 

Weight of 98 sheets, 98 X 23 . 12 = 2265 . 76 lb. 

Cost of 4000 metal slats at 8.5^ per lb. =$192.59 

Item 4- Cost of Hoops, 

Length of uppQr hoop = 56 . 07" 

Length of lower hoop = 55 . 95" 

Combined length of upper and lower hoops = 112.02" 

Total length of 500 upper and 500 lower hoops. 

Weight of 4666 ft. at 1 . 195 lb. per ft. = 5575 . 87 lb. 

Plus 5 per cent for waste =5854 . 66 lb. 

Total cost of hoops at llj?; per lb. =$658. 65 



118 • SHEET METAL DRAFTING 

Item 5. Cost of Rivets. 

Total number of 2§ lb. rivets required (500X52) =26,000 
Total number of 6 lb. rivets required (500 X 4) = 2,000 
Total number of 8 lb. rivets required (500 X 4) = 2,000 
Total number of li lb. rivets required (500X16) = 8,000 

Flat Head Tinner's Rivets are sized by their weight per thousand; i.e 
1000 rivets of 2| lb. weigh 2Hb. 

Weight of 26000-2i lb. rivets (26 X2i) = 65 lb. 
Weight of 2000-6 lb. rivets ( 2X6 ) = 12 lb. 
Weight of 2000-8 lb. rivets ( 2X8 ) = 16 lb. 
Weight of 8000-11 lb. rivets ( 8 X li) = 12 lb. 

Total weight of all rivets = 105 lb. 
Total cost of rivets at 40 f5, average price per lb. = $42 

Item 6. Drop Handles. 

Total number of handles required, 500X2 = 1000 
Weight of 36 handles and lugs, 15.66 lb. 
Weight of 1000 handles (1000^36) Xl5. 66 = 444. IS lb. 
Cost of 1000 handles at 30^ per pound = $133. 25 

Summary of Costs. 

Item 1 $491 .30 

Item 2 236.47 

Item 3 192 . 59 

Item 4 658.65 

Item 5 42.00 

Item 6 , 133.25 

Total cost $1754.26 

Cost per barrel $3.51 



CHAPTER VIII 
SECTIONS FORMED BY CUTTING PLANES 



Prob. 
No- 



JOB 



Draw \ n g 
objective 



Mathematical 
06JECTI V E 



29 




£x/e ns/on of /dec 
of cy//nc/ersa/ic/ 
corres cc// /'y 
/he //'nee/ /c^/of/es. 



CAN 



Lotera/ c?rea of 
cy//n£/er^ artd 

Vo/(//77e o/cy////dfr. 



30 




ff<7/;^a/iJ co/zs/ri/cZ/orr. 



Wt.of rods per foct 
Co/nfc///fig cosfofpu/np 



Boat Pump 



Roof Plange-S 




Cyl/nder cut by 
one p/ane.. 



/^o/re.^ 



Cose i. 




Cy//nc/er cot iy 
two p/anes. 



Cost 2 




Cy/mder cu/hy 
///ree p/o/jes . 



Ca^e 3 



Objectives of Problems on Sections Formed by Cutting Planes. 

119 



120 SHEET METAL DRAFTING 

Problem 29 
SPRINKLING CAN 

55. The Sprinkling Can. — Figure 164 represents a section of a 
sprinkling can with the names of the various parts. Special 
attention should be given to the methods of assembly. 

An elevation, Fig. 165, should be drawn according to the dimen- 
sions given in Fig. 164, 

The Spout. — The sides of the spout, Fig. 165, should be carried 
upwards until they meet at the apex. The top side of the spout, 
Fig. 165, is extended into the elevation so that the distance from the 
apex to point 1 will be equal to the distance from the apex to point 
5. The straight line 1-5 is drawn to serve as the base of the cone. 
A half-profile of the cone (spout) is drawn and divided into four 
equal spaces. Extension lines are carried from each division, 
meeting the base line at right angles. From the intersections of 
the base lines, extension lines are carried to the apex of the cone. 
The profile of the body of the sprinkling can. Fig. 166, is drawn 
next. The horizontal center line is extended to the right of the 
profile indefinitely. An extension line is dropped from the apex 
of Fig. 165 until it intersects the horizontal center line of Fig. 166. 
This will locate the apex of Fig. 166. Extension lines are dropped 
from the base of the cone (spout) in Fig. 165 until they intersect 
the horizontal center line of Fig. 166. These intersections should 
be numbered 1, 2, 3, 4, and 5 to correspond to the half-profile. 
The distances a, h, and c taken from corresponding lines in the 
half-profile of Fig. 165 are set off upon these extension lines. Lines 
are drawn from each of these points to the apex of Fig. 166. These 
lines intersect the profile of the body at three different points 
D, E, and F. Extension lines should be carried upward until D 
intersects the line from 2, F intersects the line from 3, and E 
intersects the line from 4. A curved line passing through these 
points will give the developed miter line between the body and the 
spout. 

With a radius equal to the distance from the apex to point 5, 
Fig. 165, and with the apex as a center, the arc of stretchout. Fig. 
167, is drawn. Twice as many spaces as there are in the half-pro- 
file of the spout are set off upon this arc. Lines from each of these 
points should be drawn to the apex. Straight lines are drawn 



SECTIONS FORMED BY CUTTING PLANES 121 



parallel to the base line 1-5 across the elevation of the cone, Fig. 
165, from each intersection of the developed miter line. These 
lines intersect the side or slant height of the cone at five different 
points. With the apex as a center, arcs are carried from each of 
these points over into the stretchout. Starting with point 1 of 







a-iaNVH,9.do HtaiL¥4 





cj 




u 




bO 




fl 












.^ 


»- 


a 


2 




o 


Cu 


0- 


m 


a 


1. 


5o 


o 


Z'O 


■-H 


> — 




o«> 


'■£> 


<o u. 


'"' 


u. 


m 


o 


O 






z 


I^H 




the half-profile, the extension line is traced to the base of the cone, 
thence to the developed miter line, and from this point to the 
correspondingly numbered lines of the stretchout. In like manner 
the remaining intersections may be traced. A curved line passing 
through these points will give the miter cut of the pattern. An 



122 SHEET METAL DRAFTING 

arc drawn from the apex as a center, with a radius equal to the 
distance from the apex to the top of the spout, will complete the 
pattern. A -j^-inch flange is added outside of the miter cut, as 
shown in Fig. 167, and a ^-inch lap parallel to one side of the 
spout is also added. 

Pattern of Bodij.— The pattern for the body is a rectangular 
piece of metal the length of which is equal to Diameter Xtt+I in., 
and the width of which is equal to the height of the can plus f in. 
for the wire edge, plus | in. for the single edge at the bottom of the 
can. This feature of the problem is so famihar to the student that 
it is omitted from the drawing. Manufacturers generally p-.mch 
a round hole at the point where the center line of the spout 
intersects the side of the can. The hole may, however, be devel- 
oped according to the principles laid down in Chapter V. 

The Breast. — The breast is a portion of a cylindrical surface 
cut by two inclined planes. The top hne of the breast in Fig. 165 
is extended indefinitely to the left. At right angles to this Hne 
another line is drawn to serve as a center fine for the half-profile. 
The line H-K is drawn at right angles to the center Hne. This Hne, 
if prolonged, should pass through the intersection of the breast 
and the top of the can. A distance equal to one-half the diameter 
of the can is set off upon H-K. The connecting points P and K 
of the half-profile should be drawn. The center from which this 
arc is drawn should faU on the center line of the half-profile. The 
arc is divided into four equal spaces. Extension lines are carried 
with the elevation of the breast as shown. A Hne of stretchout is 
drawn and the spacing of the half-profile with letters to correspond 
is transferred to it. From each intersection of the miter lines, 
extension fines are carried over into the stretchout. Starting from 
the half-profile, each extension line can be traced first to the miter 
fines and thence to a correspondingly lettered line in the stretchout. 
Curved lines passing through these points wiH give the half pattern 
of the breast. Fig. 168. A ^-inch lap is added to the side that 
adjoins the body, and a f-inch wire edge to the other side of the 
breast. 

The Handle. — The handle or bail of the can is a straight piece of 
metal H in. wide that is formed to the profile shown in Fig. 169. 
The handle extends down below the top of the can and is riveted 
and soldered to the body. A double hem or a wire may be used to 
stiffen each edge of the handle. A boss is soldered into the upper 



SECTIONS FORMED BY CUTTING PLANES 123 

part of the handle to aid the hand in giipping it. The handle does 
not require a pattern since it is a straight strip of metal whose 
length is equal to the perimeter of the profile, Fig. 169, and whose 
width is equal to 1 j in. plus the allowances for stiffening each side. 
The pattern of the boss is obtained in exactly the same manner as 
the pattern of the breast, and needs no further description. Figure 
170 shows the profile of the S handle together with its boss. 
The pattern for the S handle is a tapering strip of metal whose 
length is equal to the distance around the profile, Fig. 170, and 
whose large end is Ij in. wide, while the small end is f in. wide. 
A j-inch wire edge is added to each side for the stiffening wire. 

56. Related Mathematics on Sprinkling Can. — Volume of 
Sprinkling Can. — In Chapter II it was learned that the volume of 
a cylinder is equal to the area of the base multiplied by the height, 
or V=^AxH. It was also found that the area of a circle was equal 
to the diameter squared, multiplied by .7854, or D^ X . 7854. 

Sprinkling cans are generally listed according to their holding 
capacity in gallons, quarts, and pints. 

Sample Problem. — What is the capacity of a sprinkling can whose diameter 
is 8" and whose height is 10|"? 

Formula: y = D2X.7854Xff 

Substituting, V = 8^ X . 7854 X 10 . 5. 

F = 64X. 7854X10. 5. 
.7854 
64 



31416 
. 47124 

50.2656 
10.5 

2513280 
5026560 

527.78880 cu. in. 
There are 231 cu. in. in one gallon; therefore, 
231 I 527 . 789 |2.28 gallons 
462 



657 
462 

1958 

1848 



1109 Ans. 2.28 gal. or 9 qt. 

But, 2.28 gallons is approximately 9 quarts, the extra contents being 
allowed for carrying nine full quarts without danger of spilling. 



124 SHEET METAL DRAFTING 

Problem S9 A. — What is the capacity, m quarts, of the follow- 
ing standard sizes of sprinkling cans: 

Diameter of Bottom. Height of Body, 
(a) 4" 41" 

5'f 7 3 / 



(6) 51" ■ 7t 



(c) 71" 9" 

(d) SW' lli^" 

Problem ^P5.— Regarding the body blank for the can as a 
rectangle whose height is equal to 10|"+f"+i^" and whose 
length is (8X3. 1416) +|" what is its area in square inches? 

Problem 29C. — Regarding the bottom of the can as a circle 
whose diameter is equal to 8" + |"+f", what is its area in square 
inches? 

Problem 29D. — Regarding the whole pattern of the breast, 
Fig. 168, as the combined area of two right triangles, having bases 
5^" long and altitudes 8" high, what is the total area of the breast? 

Problem .g^^J.— Regarding the spout as a trapezoid whose upper 
base is If", whose lower base is 9", and whose altitude is \^" , 
what is its area in square inches? 

Problem 29F. — Regarding the semicircular handle as a rec- 

tangular piece of metal whose length is ^^+4", and whose width 

is l\" + \" + \", what is its total area? 

Problem ^5G.— Regarding the "boss" in Fig. 169 as a rectangle 
whose length is 4|", and whose width is 2", what is its total area? 

Problem 29H. — Regarding the S handle as a trapezoid whose 
upper base is f", whose lower base is l\", and whose altitude is 
b\", what is its total area? 

Problem ^57.— Regarding the "boss" in Fig. 170 as a rectangle 
whose width is Ir^" and whose length is 2\" , what is its area? 

Problem 29J. — Adding together the answers to problems, 
29B, C, D, E, F, G, H, and I, what is the combined area of all the 
parts? 

Problem 29K.—A sheet of IXXXX charcoal tin measuring 
20"X28", costs 50 cents. Adding 5 per cent to the answer of 
Problem 29J, how much will the stock required for one sprinkling 
. can cost? 



SECTIONS FORMED BY CUTTING PLANES 125 

Problem 30 
BOAT PUMP 

57. The Boat Pump. — A boat pump consists of a straight piece of 
pipe called the barrel, to which is attached a frustum of a cone (the 
funnel). A tapering spout is riveted and soldered over an open- 
ing in the pump barrel. Into the lower end of the pump barrel a 
''lower box" is soldered. The "lower and upper boxes" may be 
obtained from almost any supply house. The upper box is 
threaded to receive the pump rod. The pump rod has an oval 
handle formed upon its upper end. Measurements are usually 
given from the under side of the spout to the lower end of the 
pump, for the diameter of the barrel, and for the length of the 
spout. The other details of construction are left to the discretion 
of the designer. 

The Barrel. — The barrel of the pump is made from one piece 
of metal, if possible, in order to avoid any possibility of the upper 
box catching as it works up and down in the cylinder. The 
pattern is a rectangular piece of metal the length of which is shown 
in the elevation, Fig. 171, and the width of which is equal to 
(Diameter X7r) + ^ in. for locks. 

The Spout. — An elevation should be drawn according to the 
dimensions given in Fig. 171. The center line of the spout is 
next drawn and prolonged to the right indefinitely. The sides of 
the spout are extended until they meet the center line at the apex. 
They are also extended to the left until they meet the center line 
of the barrel. This will give a right cone whose base is the line 
1-5. A half-profile for this cone should be drawn. It should be 
divided into four equal parts, and each division numbered. A 
profile of the barrel. Fig. 172, should next be drawn. Using the 
same center, a half-profile of the spout. Fig. 172, should be put in. 
A horizontal center line that is long enough to receive an extension 
line dropped from the apex of Fig. 171 should also be drawn. This 
half-profile is divided into four equal parts and these divisions 
numbered to correspond to those of the half-profile in Fig. 171. 
These numbers change their positions as can be seen. From each 
division in both half-profiles, lines are drawn to the bases of the 
cones forming angles of 90°. From each intersection thus found, 
lines are drawn to the apex. In Fig. 172, the line 1 intersects the 



126 



SHEET METAL DRAFTING 



profile of the barrel at point a, line 2 at point b, and line 3 at 
point c. Extension lines are carried up into Fig. 171 from points 
a, h, and c of Fig. 172. At the points where these lines intersect 




Figs. 171-175.— Boat Pump. 

corresponding lines in Fig. 171, will be the location of the new 
points a, b, and c of Fig. 171. A curved fine traced through these 
points will give the developed miter line. 



SECTIONS FORMED BY CUTTING PLANES 127 

Pattern of Spout. — The arc of stretchout, Fig. 173, is drawn with 
a radius equal to the slant height of the cone, and with the apex as 
a center. The spacing of the half-profile is transferred to this arc 
with numbers to correspond. Where extension lines from points 
a, h, and c cut the slant height of the elevation, extension arcs are 
drawn over into the stretchout. All intersections should be traced 
out by starting from the profile, following each extension line to 
the miter line, and thence to a correspondingly numbered line in 
the stretchout. The miter cut of the pattern is obtained by trac- 
ing a curved line through these intersections. An arc whose 
radius is equal to the distance from the apex to the end of the 
spout completes the pattern. A lock on each side of the spout 
and a hem on the small end of the spout are added. A flange 
(not shown) should be added to the miter cut. 

The Opening. — Upon any straight line, the distances a to 6 
and 6 to c of Fig. 172 are set off. Since this is but half of the open- 
ing, this operation must be repeated as shown in Fig. 174. Meas- 
uring lines are drawn through each of these points. Upon the 
lines h the distance from point 2 to the center line of the half- 
profile of Fig. 172 is set off. Upon the lines a, the distance 
from point 1 to the center line of the half-profile of Fig. 172 is set 
off. Points c fall upon the center line of Fig. 174. A curve 
traced through the points thus obtained will give the shape of the 
opening. 

The Funnel. — One side of the funnel should be extended inward 
until it meets the center line of the barrel. This will locate the 
apex of the whole cone, of which the frustum is a part. With any 
convenient point as a center, and a radius equal to the distance 
from this apex to the large end of the funnel, an arc of stretchout. 
Fig. 175, is drawn. A quarter-profile is placed above the elevation 
of the funnel in Fig. 171. This is divided into three equal parts. 
Since this is but a quarter-profile, twelve spaces must be trans- 
ferred to the arc of stretchout in Fig. 175. The first and last points 
are connected to the apex by means of straight lines. The pattern 
is completed by an arc, drawn from the center of Fig. 175, whose 
radius is equal to the distance from the apex of the funnel to the 
point F in Fig. 171. The necessary locks and wire edge should be 
added. 

58. Related Mathematics on Boat Pump. — Problem 30 A. — The 
barrel of the boat pump is a cylinder whose diameter is 4" and 



128 SHEET METAL DRAFTING 

whose height is 5' — 0". Allowing |" for locks what is the area 
of its pattern? 

Problem SOB. — The funnel of the pump is a frustum of a cone 
whose lower base diameter is 8", upper base diameter 4", and 
altitude 4". What is its area? 

Sample Problem. — Substituting for the above dimensions 6" upper base, 
3" lower base, and 3" altitude, we would have 
Original Formula : 

Circumference of upper base+Circumference of lower base. 



X slant height. 



Substituting, 



(6X 5)+(3X.) ^^3,^^ 



The slant height of any frustum of a right cone is equg,! to the hypotenuse 
of a right triangle whose base is the difference between the radius of the upper 
base and the radius of the lower base, and whose altitude is the altitude of the 
frustum. The slant height of this frustum is, therefore, equal to the hypote- 
nuse of a right triangle whose base is 3" — 15" = H", and whose altitude is 3". 
But the hypotenuse is equal to Vbase^+altitude-; therefore, the slant height 
for this frustum is V32-I-I.52 

Solving further, o..o-t^ .^^ X ^^9 +2 . 25 = 

14.13X3.35 = 47.33 sq.m. 

Problem 30C. — The spout, Fig. 171, is also a frustum of a cone 
whose lower base has a diameter of 3|", and upper base a diameter 
of 2|". The slant height may be called 7|". What is its area? 

Problem SOD. — What is the combined area of Problems 30A, 
SOB, and 30C? 

Problem 30E. — Allowing 10 per cent for waste, what will be 
the cost of the material for one pump at 8 cents per square foot? 



SECTIONS FORMED BY CUTTING PLANES 129 

Problem 31 
ROOF FLANGE 

59. The Roof Flange. — Case I. The Roof Having One Inclina- 
tion. — The measurements usually given for a roof flange are the 
diameter of the pipe, and the pitch of the roof. The roof pitch 
is generally given as "so many inches to the foot." Figure 176 
shows the rise and run of the roof line. If the job called for a roof 
pitch of 4 in. to 1 ft., the line marked run, Fig. 176, would meas- 
ure 12 in. and the line marked rise would measure 4 in. 

Pattern of Pipe. — An elevation is first drawn according to the 
dimensions given in Fig. 176. A profile is drawn above this view 
and is divided into equal spaces. Each division is numbered. 
Extension lines from each division of the profile are carried down 
to the roof line. A line of stretchout is drawn, and the spacing 
from the profile transferred to this line. The spaces are numbered 
to correspond. The measuring lines of the stretchout are drawn 
in. From each intersection of the miter (roof) line of Fig. 176, 
extension lines are carried over into the stretchout. Starting 
from point 1 of the profile, the extension lines should be traced 
downward to the miter line, and thence to a correspondingly 
numbered line in the stretchout, Fig. 177. In like manner all 
points of intersection can be located in the stretchout. A curved 
line passing through these points will give the miter cut of the 
pattern. An extension line drawn from the top of the roof flange 
elevation completes the pattern. A ^-inch lock is added to each 
side of the pattern and a f-inch double edge to the miter cut in 
order to join the pipe to the apron by double seaming. 

Opening in Apron. — Any straight line is drawn to serve as a 
Hne of stretchout, Fig. 178. The exact spacing between the inter- 
sections of the miter line is transferred to this line, and these spac- 
ings numbered to correspond. A measuring line is drawn through 
each point as shown in Fig. 178. Upon line 2 of Fig. 178, a distance 
equal to hne a of Fig. 176 is set off. Similarly, line 3 would 
receive the length of Hne h from Fig. 176, line 4 would receive c, 
line 5 would receive d, and line 6 would receive e. These distances 
may now be transferred to the opposite side of the line of 
stretchout, since both parts of the opening are exactly equal. A 
curved line drawn through the points thus located will give the 



130 



SHEET METAL DRAFTING 



shape of the opening. The apron of the flange may now be drawn 
around the opening, allowing at least 3 in. on the sides and bottom 



fHem 




nc. 180 
Figs. 176-181.— Roof Flanges. 



and 6 in. on the top. A hem should be added to the long sides of 
the api-on to direct the flow of water. 



SECTIONS FORMED BY CUTTING PLANES 131 

Case 11. The Flange Fitting over the Ridge of a i?oo/.— Figure 
179 shows an elevation and profile of a roof flange fitting over the 
ridge of a 90° or "square-pitch" roof. Figure 180 shows the pat- 
tern for the pipe. It will be noticed that this type of flange can- 
not be easily double seamed; therefore, a ^-inch edge is added 
to the miter cut. This edge is turned off, and riveted and 
soldered to the apron. Figure 181 shows the pattern of the apron. 
A bend of 90° must be made on line 4 in order to fit the apron 
over the ridge of the roof. The description for Case I will apply 
to this problem, the only difference being the shape of the miter 
cut in Fig. 176. 

Case III. The Flange Fitting over the Ridge and Hips of a 
Roof. — An elevation is first drawn according to the dimensions 
given in Fig. 182. Above this elevation a half-profile is drawn and 
divided into equal spaces and numbered as shown. Extension 
lines are carried from each division down to the roof line. A plan. 
Fig. 183, is drawn in the following manner: Extend the center line 
of the elevation downward indefinitely. Construct a rectangle. 
Fig. 183, using this line as a center line. This rectangle will repre- 
sent the top view or plan of the apron of the finished flange. 
Draw two lines at an angle of 45° to represent the hips of the roof. 
The center line becomes the ridge. With the point where the 
ridge and hips meet, as a center, and a radius equal to that of the 
half-profile. Fig. 182, draw a circle. Divide this circle into twice 
as many equal parts as there are in the half-profile. Number 
these divisions to correspond. The hips cross this circle halfway 
between points 2 and 3 and 5 and 6. Number these points 2| 
and 5| respectively. Carry extension lines upward from each 
division of the circle, to the roof line. 

That part of the miter line not already shown in elevation can 
be developed. Points 2| and 5^ in Fig. 182 occur where the 
lines from these points in Fig. 183 intersect the roof line in Fig. 182. 
Since this is the highest point of the miter line on the hips, point 3 
must.be opposite point 2, point 4 opposite point 1, and point 5 
opposite point 6. This is indicated by horizontal dotted lines in 
Fig. 182. A curved line drawn through these points will be the 
developed miter line. A line of stretchout, Fig. 184, is now 
drawn. The exact spacing is transferred from the circle in Fig. 
183 to this line, and numbered to correspond. 

The measuring lines of the stretchout are drawn. Starting at 



132 



SHEET METAL DRAFTING 



point 1 of the plan, the extension hne is traced to the miter line, and 
thence to a correspondingly numbered line in the stretchout. In 
like manner, all points of intersection can be located in the stretch- 
out. A curved line passing through these points will give the miter 
cut. The pattern is completed by an extension line drawn from the 



m 





.^<r^ 


0> N. 


/ « 


. \ 


I - 






V. 


vv 


* // 1 


\ rfs 


'H// 


"A 


r^ 


U" 


N) 




/ 


> 




^ 



a, 
W 

PI 



Ph 



f^ 



f^ 



P4 



top of the pipe. A lock is added to each side of the pattern. A 
flange (not shown) should be added to the miter cut. 

The Apron. — The exact spacing of points 10, 11, 12, 1, 2, and 
2|, as shown on the center line of Fig. 183 by lines a, b, c, d, and 
e, are set off upon any straight hne, Fig. 185. These points also 



SECTIONS FORMED BY CUTTING PLANES 133 

bear the numbers 9, 8, 7, 6, and 5| as shown in Fig. 185. Measur- 
ing Unes are drawn through each of these points. The distances 
a, b, c, etc., are taken from hnes, a, h, c, d, and e of Fig. 183 and are 
set off on the measuring hne. A curved hne passing through 
these points will give the shape of the opening for the main part of 
the flange that is to fit over the ridge. The distance 10-ikf, Fig. 
185, is made equal to 10-ilf of Fig. 183. A perpendicular is drawn 
at point M. The distance MN is made equal to 10-7V of Fig. 182. 
Another perpendicular line is erected at point N. Distance 
NF is made equal to NF of Fig. 183. The bending line FG 
is drawn in. With F as a, center, arcs are drawn from points 
H, J, and K to the left an indefinite distance. With G as a center, 
the distance GH is set off on the other side of point G. Similarly, 
the distances GJ and GK are set off on the other side of point G. 
The straight lines KF and KJ will complete the pattern of one 
side of the flange, with the exception of the curve JHG. The 
other half of the pattern is exactly equal to the one already drawn 
and is produced by the same method. 



CHAPTER IX 
FRUSTUMS OF RECTANGULAR PYRAMIDS 



Pp>b. 
No. 



JOB 



DBJE.CTIVI 



Hathematical 

JECT IV E. 



Zt 




Dripping Pans 



/e/ea of fire patf 
corner 5 fOfrdart:/ 
ru/P for j//e o//>affj. 



SobfraeH on of 
fractions. 



J3 



Range Pans — 
Unequal Flare 



V/€¥fS necessary 
to ^/ve cc^f/efff 

fnformafio/t. 



Area of a 
trapezo'fd. 



Recii ste-h 6ox 



34 




consfrc/c//'o/7. 



free drea of 
re^/sfer ^^^ 
e^c//Va/e/7f ftps 
ffzes. 



Objectives of Problems on Frustums of Rectangular Pyramids. 

135 



136 SHEET METAL DRAFTING 

Problem 32 
DRIPPING OR ROASTING PAN 

60. Dripping or Roasting Pan. — ^Figure 186 is a sectional view 
of a standard dripping or roasting pan. Since pans of this kind are 
subjected to temperatures above that at which solder melts, it is 
necessary to employ some means besides soldering to enable them 
to hold liquids. This is accomplished by folding the corners of the 
pattern so that they will lie flat against the ends. 

Standards of Construction. — All dripping pans have a standard 
flare of three-eighths of an inch as shown in Fig. 186. They are 
wired with No. 8 tinned or coppered wire. The measurements 
for this type of pan are always understood to be measurements of 
the top of the pan outside of the wire. The dimensions of the 
bottom are always IJ in. less than the measurements of the top. 
Thus, a 12 in. X 18 in. dripping pan measures 12 in. X 18 in. outside 
the wire around the top, and the bottom measures lOf in. Xl6f in. 
The depth varies with the size of the pans. 

Pattern of One Corner. — Figure 187 shows a full size develop- 
ment of one corner of a dripping pan, which is produced in the fol- 
lowing manner: 

Draw the lines ah and he forming an angle of 90°, Fig. 187. 

Continue these hues indefinitely (shown by dotted lines hd 
and he), and set off upon these lines the slant height of the pan, 
hd of Fig. 187. 

Draw lines dH and ek parallel to lines ha and he, respectively. 

Make lines df and eg three-eighths of an inch long. 

Prolong these lines until they meet at the point L. 

Draw the diagonal Lh. 

With * point / as a center, and any radius, draw the arc 
mnp. 

With n as a center, set off the distance mn on the other side 
of the point n so that arc np will exactly equal arc 7im. 

Draw the line fg, extending it until it intersects the diagonal 
at the point r. 

Draw the line rg. 

Draw the lines s~T and t-u parallel to/-r and i^-g, and at a dis- 
tance of I in. 

This cutting away of the corner allows the wire around the top 



FRUSTUMS OF RECTANGULAR PYRAMIDS 137 

of the pan to lie flat against the ends and prevents thereby a 
"bunch" at each corner. 

A wire edge of f in. should be added to each top edge as shown 
in Fig. 186. 




FiR5T Operation Second Operation Thirp Operation Fourth Operation 



.XL. 



14.- 




.1 s Wire Edqe^^ 




I4 Less than oufside mcasui-ementa 



S lani hclM 




Fig. 188 



1 




rx 

<oico Lay-out of Pan on Metai_ Bcank 

Figs. 186-188. — Dripping or Roasting Pan. 

First Operation. — The outline of the bottom of the pan is swaged 
on all sides by means of the " Square Pan Swage." The diagonals 
(hne h. T. of Fig. 187) are next swaged in the same machine. 

Second Operation. — The sides are "brought up" simultaneously 



138 SHEET METAL DRAFTING 

over the hatchet stake, the corners taking the shape shown in the 
illustration. These corners are then closed down on the "square 
head." 

Third Operation. — Each corner in turn is then placed in posi- 
tion on the "crooked square head" and the corner "brought 
around" as shown in the illustration. 

Fourth Operation. — The wire edge is then "laid off" over the 
edge of the bench, the wire inserted, and finished in the wiring 
machine. The bottom edges are then straightened and the oval 
handles attached to the ends of the pan. 

Laying out Directly upon the Metal. — Figure 188 shows the pat- 
tern of a dripping pan measuring 12f in. X5f in.Xl in. deep. 
The first step is to compute the size of the blank required. Since 
the bottom is Ij in. smaller than the top, the bottom measure- 
ments would be 

12r-li"=ll|" (length) 

5|"_li"= 4|" (width) 

To each dimension of the bottom must be added two slant 
heights and two wire edges. The slant height of a pan 1 in. deep 
and f in. flare is 1^^ in.; therefore, the dimensions of the blank 
would be 

lir+(li^X2) + (fX2) = 14" (length) 

4r+(liVX2) + (fX2)= 7" (width) 

The workman cuts a rectangular piece of metal 7 in. X 14 in. 
and lays off a f-inch wire edge, Fig. 188. Inside of these lines he 
lays off a distance of Ire in. to form the outline of the bottom. He 
then develops the pattern of one corner, according to the description 
given for Fig. 187. Having obtained the pattern for one corner, he 
transfers the measurements to the other three corners and the 
pattern is complete. The shaded portion of Fig. 188 denotes that 
part of the blank which is cut away. 

61. Related Mathematics on Dripping Pans. — Dripping pans 
are made in a great variety of sizes, the more common of which 
are: 10"X15"X2|"; 12"X17"X2J"; 12"Xl9"X2i"; 14"X 
15"X2i"; and 18" X 19" X 21". 

Problem 32 A. — What will be the dimensions of the blanks 
required for each of the above sizes of pans? 



FRUSTUMS OF RECTANGULAR PYRAMIDS 139 

Problem 32B. — What will be the weight of each of the above 
sizes of pans if made from No. 28 U. S. S. Gage black iron ( . 6375 
lb. per sq. ft.) and "wired" with No. 8 gage wire weighing .07 lb. 
per running foot? (Do not deduct for the corners that are cut 
away.) 

Problem 32C. — Using stock size sheets of 30"X96" black iron, 
what would be the percentage of waste per pan for dripping pans 
measuring 12"X17" and 2\" deep? 



140 SHEET METAL DRAFTING 

Problem 33 
RECTANGULAR FLARING PANS 

62. Rectangular Flaring Pans. — Frequently, the contour of the 
ash pit demands that a furnace or range ash pan be given unequal 
flares. Such a pan is shown in Figs. 189 and 190. It should be 
noted that the sides of the pan flare 2 in. as shown in Fig. 189, 
while the front end flares 5 in., and the back end flares 3 in. as 
shown in Fig. 190. 

The Elevation. — A front and side elevation should be drawn, 
setting forth the exact dimensions of the pan. The pan is stiffened 
around the upper edge with a |-inch rod. When this rod is covered 
with the wire edge, not less than f in. should be allowed for clear- 
ance. Thus a pan measuring 34| in. X 16f in. outside of the wire 
would measure 34 in. X16 in. inside of the wire. The pan has 
two profiles A, B, C, D of Fig. 190, and E, F, G, H of Fig. 189. 

The Pattern. — Two Hues of stretchout are drawn at right angles 
to each other as shown by the dotted Hues of Fig. 191. The spac- 
ings of the profiles are set off upon these fines of stretchout, the 
spacings A,B,C, and D being taken from Fig. 190, and the spacings 
E, F, G, and H being taken from Fig. 189. Measuring lines are 
drawn at right angles to the fine of stretchout through each of these 
points. The flares 5 in., 3 in., and 2 in. should now be set off at 
their respective corners as shown in Fig. 191. The inclined lines 
representing the meeting fines of each corner should be put in. 

A |-inch lap is added to each corner of the sides of the pan, and 
a center line for the rivet holes drawn in. The centers for the rivet 
holes should also be located in the laps. With the corner of the 
bottom (point P) as a center, arcs intersecting a straight line 
drawn f in. in from the meeting line of the end should be drawn. 
Only one corner of Fig. 191 is treated for rivet holes, but it is re- 
quired that the rivet holes for afi corners be located. The f-inch 
edge for covering the j-inch rod may now be added to complete 
the pattern. 

The Bail— -The bail is shown in Figs. 189 and 190 by dotted 
fines. Since the bail is below the top fine of the pan, it must be 
drawn carefully in order to scale the dimensions accurately. It 
is generally placed so that the bail ear will be to the rear of the 
center. The ear is attached in such a manner that the" stop" will 



FRUSTUMS OF RECTANGULAR PYRAMIDS 141 

maintain the bail in an upright position. The center of gravity 
being in front of the bail allows the pan to be carried with one hand 
without danger of spilling the contents. 



I H« - lg" " IE A|^ 



-34"- 



^a'V 



\z 



End Elevation 



-H2> 



V — -Bail 3" rod 



Fig. 190 



4" Rod 



■5"->-}*- 



Z6'- 



-H 3" 



5lOE EJ.EVATION 



1;:: 



5"-H— 



ae"- 



"T 




wire edge 



"H*"'" I— — 



Whole Patitcrn of Pan 

I Punchfor2r 




jii: 



Elevation of Bail 

Fig. 19a 



Jib 




Figs. 189-192. — Rectangular Flaring Pan. 



Figure 192 shows an elevation of the bail. The dimensions are 
all given for the center hnes of the rod. In case the rod is bent to 
a "close angle" in the vise, these dimensions will answer. How- 
ever, if a long radius bend is desired, the actual length of the center 



142 SHEET METAL DRAFTING 

line must be computed, in order to obtain the length of the blank 
piece of rod required to make the bail. 

63. Related Mathematics on Rectangular Flaring Pans. — 

Problem 33 A. — How much will the flaring pan of Fig. 191 weigh 
if made from No. 20 U. S. S. Gage black iron (1 .53 lb. per sq. ft.)? 
The pattern is to be regarded as a rectangle. 

Problem 33B. — How much will the rod around the top edge of 
the pan, Figs. 189 and 190, and the rod required to make the bail, 
Fig. 192, weigh if \" rods weigh 0.1669 lb. per ft. 

Problem 33C. — How much will the material required to make 
the complete pan cost at 7 cents ner pound? 



FRUSTUMS OF RECTANGULAR PYRAMIDS 143 



Problem 34 
REGISTER BOXES 

64. Register Boxes. — Register boxes are generally made from 
1 C coke tin, commonly called by the trade "furnace pipe tin." 
The tin box must be made to fit the body size of the register. An 
allowance is generally made to assure an easy "fit" between the 
body of the register and the box. This allowance varies with the 
sizes of the registers as follows: 



Size of Register Body. 


Dimension of Box. 


Depth of 
Box. 


Not wider than 4 in., any length 


Add J in. to each dimension 






of body 


4 in. 


Not wider than 5 in., any length 


Add j^ in. to each dimension 






of body 


4 in. 


Not wider than 7 in., any length 


Add j^ in. to each dimension 






of body 


4 in. 


Not wider than 8 in., any length 


Add f in. to each dimension 






of body 


4 in. 


Not wider than 11 in., any length 


Add xi in. to each dimension 






of body 


5 in. 


Not wider than 12 in., any length 


Add 1 in. to each dimension 






of body 


5 in. 


Not wider than 18 in., any length 


Add 1 in. to each dimension 






of body 


6 in. 


Wider than 18 in., any length 


Add xf "!• to each dimension 






of body 


6 in. to 8 in. 



Figures 193 and 194 show end and side elevations of a register 
box for a 9 in. X12 in. register. According to the table, the 
dimensions of the top will be 9^^ in.Xl2x^ in. 

End Elevation. — The end elevation is drawn according to the 
dimensions given in Fig. 193. A half -profile of the "neck" is 
drawn and divided into equal spaces and each space numbered. 
The neck is joined to the box by a "bead and flange" joint. The 
corners of the elevation are numbered 1, ^, 2, 3, .S, and 4 as shown. 

Side Elevation. — A side elevation should be drawn accord- 
ing to the dimensions given in Fig. 194. The points 5, C, 6, 7, D, 
and 8 are numbered as shown. 

Pattern of Ends. — First, any horizontal straight line (line 2-3 of 



144 



SHEET METAL DRAFTING 



Fig. 195) is drawn equal in length to line 2-3 of Fig. 193. A per- 
pendicular is erected at the point 2. The length of the slant height 
(Hne c-6 of Fig. 194) is set off from tliis perpendicular. The line 




it-- 




il iwo wanTud 
Fig. 195 




gSingle edge 

Pattern of Ends 
(Two wan+«.d) 



2 Double edge FiG. (96 



R<NTTE.RN OP SrDES 

(TWo waatad) 

§' Single edge 7; 







i 



" "i^ "IS \6 'n 16 

Pattern of Neck 
Fig. 197 



IS !■» 13 '12 



z locK- 





FiGS. 193-198.— Register Boxes. 



A B is drawn parallel to line 2-3. The point A is located f in. 
distance from the perpendicular. A |-inch single edge is added to 
edges 4-2, 2-3, and 3-B, and a |-inch flange is added to the top 
edge of the pattern, mitering the corners at an angle of 45°. 



FRUSTUMS OF RECTANGULAR PYRAMIDS 145 



Pattern of Sides. — The pattern of the sides, Fig. 196, is produced 
in like manner. Instead of the single edge, however, a double edge 
is added to the edges that are to double seam onto the ends. 

Pattern of Neck. — A hne of stretchout, Fig. 197, is drawn and 
upon it is laid off twice as many spaces as there are in the half- 
profile of Fig. 193. Perpendiculars are erected at the points 9 
and 9, Fig. 197. One-fourth inch edges are added to each end 
for the standard tin lock, and a j^-inch edge for the bead and 
flange joint notching as shown in Fig. 197. 

Pattern of Bottom. — A rectangle. Fig. 198, whose dimensions 
are f in. less than the top dimensions of the box, is drawn. The 
center is located by drawing the diagonals of this rectangle. From 
this center, a circle whose diameter is | in. less than the diameter of 
the neck, is drawn as shown in Fig. 198. This circle is cut out of 
the metal to provide an opening for the neck, and is always made 
smaller because the bead "draws in" when it is turned in the thick 
edge. A J-inch double edge is allowed on all sides to provide for 
double seaming the bottom to the body of the box. Over-all 
dimensions are placed on all views. 

65. Related Mathematics on Register Boxes. — Problem 34A. — 
Furnace pipe tin is made in the sizes listed in the following table. 
Which size would you use in making the register box shown 
by Figs. 193 to 198 inclusive, in order to maintain as httle waste 
as possible? 

Coke Tin — Furnace Pipe Sizes 



Size of Sheet. 


Wt. Per Box 

No. Sheets. 


Corresponding 
Pipe Size. 


20"X23" 


165 lb. 


7" 


20"X26|" 


190 " 


8" 


20"X29i" 


211 " 


9" 


20"X32i" 


233 " 


10" 


20"X36" 


258 " 


11" 


20"X39" 


290 " 


12" 



Problem 34B. — Any register has a series of holes cast in its face 
to correspond to some predetermined design. This design neces- 
sarily shuts off part of the opening, thereby retarding the flow of 
air through the register. Since most makers use nearly the same 
design, it has become the custom to deduct 33§ per cent of the area 



146 



SHEET METAL DRAFTING 



of the body size, in order to obtain the free area of the register. 
Fill in, in the following table, the free areas of the register sizes 
given. 

Free Areas of Registers 





Free Area 


Size of Round Pipe 


Size of Body. 


(66f per cent ot 


Required (See Prob- 




body area). 


lem 34C). 


6"X10" 






8"X12" 






9"X12" 






9"X14" 






10"X12" 






12"X14" 






12"X16" 







Problem 34C. — Since a register cannot deliver more air than is 
conveyed to it, it is evident that the cross-sectional area of the 
round neck of the register box must equal the free area of the regis- 
ter. Using the formula, D== V Free area-^ .7854, fill in the third 
column of the above table. 



CHAPTER X 
COMBINATIONS OF VARIOUS SOLIDS 



Prob. 

Ho. 



Job 



Drawin g 
Objective. 



Mathematical 
Objective 



35 




ATOMIZ.ING 
SPKAYE.W. 



36 



Ash Pan 

(Round end) 



AssemMy draw'im 
end devdopmnr 



4 ma afcy/Merj. 




(omtinafm of 
f/af and cwKed 
fi/rfaces. 



4r^a offrapezo/d, 
fraxtum afco/ie, 
a/7^ of c/rc/e. 




AnemtJy drawing. 
Pe¥elopme/ff of 
paff^r/j.Pefa/fs 
of co/rjfr(/£fion. 



Compt/f//r^ ^jrpas 
of i^ari(H/s parts. 



Objectives of Problems on Combinations of Various Solids. 
147 



148 SHEET METAL DRAFTING 

Problem 35 
ATOMIZING SPRAYER 

66. The Atomizing Sprayer. — This type of sprayer consists of a 
cylindrical tank or reservoir in wliich is soldered a small tube that 
reaches nearly to the bottom of the tank. Upon this tank is 
mounted a pump, having a nozzle in the form of a scalene cone, 
and with a relatively small orifice. The small tube is placed in the 
tank in such a manner that its top is directly in front of, and on a 
level with, the center of the orifice of the pump. When a stream 
of air is expelled from the pump, it creates a partial vacuum in the 
tube, causing the Hquid to rise. When the hquid encounters the 
stream of air flowing from the pump, it is broken up into a fine 
spray. 

The Elevation (Fig. 199). — A circle 4 in. in diameter is first 
drawn to represent the end of the reservoir. The elevation of the 
pump barrel is drawn next, placing it in a horizontal position and 
tangent to the reservoir. The following are next drawn in order: 
(1) The pump rod assembly; (2) the j^^-inch diameter tube in its 
proper location; (3) the wooden plug that guides the pump rod; 
(4) the brace, according to. the dimensions given. 

Pattern of Conical Nozzle (Fig. 200). — The elevation of the 
cone is reproduced and a half-profile attached directly to the base. 
This half-profile is divided into four equal parts. The shortest 
distance from the base to the apex is the line from point 1. There- 
fore, with point 1 as a center, arcs are drawn from points 2, 3, and 
4 of the half -profile, cutting the base of the cone as shown in Fig. 
200. With the apex as a center, arcs are drawn from each inter- 
section of the base of the cone. Any point on the arc from point 1 
is selected and is connected with the apex. This line will serve 
as the starting line of the pattern. The compass is set equal to the 
distance between any two divisions of the half-profile. Starting 
from point 1 of the pattern, point 2 is found by drawing an arc 
(with the compass as already set) that intersects the arc drawn from 
the second intersection of the base line of the cone. In like man- 
ner points 3, 4, and 5 can be located. A straight line from point 5 
to the apex will give the half pattern. Since both halves are 
exactly equal, the other half may now be drawn by reversing the 
process already described. A j-inch lap should be added to one 
side of the pattern. 



COMBINATIONS OF VARIOUS SOLIDS 



149 



.Iron Washer *2 





Reservoir 4" Diam.X5''Long 



4 Pump Rod' 
Hondle' 



Elevation of Atomizing Spray Pump 

Fie. 199 




=■5 



/locK-^ 



Drill i 
Drill ^•■ 

Pattern of | ) Pump Barrel 
FigI. /goi 



:^x- 




i -16 S.B.Thd. 



Stop Washer 



Pattern of Cone 
Fig. ZOO 



ki'M^ 



i/i- 



^ 



3EHiH33> 



Pump Rod 



/Drill +Tapi"-iaThds.v ,» 



/Drill +Tapi"-iaThds.v ,« 

' - '^^ ^ ,,. „ Drill i N 

. r„ i, #, /ron washer «. Cup L.oth.r 5top Wosher 

Iron Washer *2 



Pump Rod and Packing Details Fig. ZOZ 



VA 



o Punch 

c ~ , ~ 

= Punch for 2 Screw anTop 

Pattern for Boov of Reservoir 



^ 



S / 

-■ T) 

I- «) 

y s 



i« 




i"Burr Fig. 203 

Pattern for 
Ends of Reservoir 



6CHEDULE OF MATERIALS || 


Item 


Detail No 


No.Reou 


RID 


M/niKiAi. 


Cone 




12 


ECbar.Tm 


Pump Barrel 




IE 




Iron Wa5)icr*2 




IZ 


llGamcTrriK, 


Iron Washer* 1 




IZ 


■' 


5top Washer 




12 




Cup Leather 




IZ 


stock Room 


Pump Rod 




12 


Golv.RodCi", 


Dodyof Rtservoir 




IZ 


KChor.Tin 


Brace 




12 




Ends of Reservoir 




24 




Handles 




!2 


Stock Room 


Pluqs (Maple) 




IZ 


•• 










Regul 


^R 


T 


TLE 


H 


ERE 



Figs. 199-203. — Atomizing Spray Pump. 



150 SHEET METAL DRAFTING 

Pattern of Pump Barrel (Fig. 201).— The pump barrel blank 
is a rectangular piece of metal whose length is 19f in. (| in. 
being added for joining to the nozzle), and whose width is 
(1| in. Xtt) + ^ in. for locks. A pattern of the pump barrel should 
be drawn as shown in Fig. 201, The |-inch screw holes should be 
equally spaced with the circumference of the barrel, the outside 
holes being | of the circumference distant from the circumference 
lines, in order to bring the seam in the center. A j^-inch hole 
must be provided, as shown, for a vent. 

Pump Rod Details (Fig. 202).— The pump rod should be 21| in. 
long and should have a stop washer soldered 3| in. from one end. 
This stop washer prevents the leather packing from becoming 
injured by striking the nozzle. A |-inch standard stove bolt thread 
(18 threads per inch) is cut on each end of the rod for a distance of 
I in. The thread near the stop washer is intended to screw into 
wooden handle. Two iron washers of unequal diameter are 
drilled and tapped to receive the thread that is cut on the pump 
rod. The cup leather is clamped between these washers, the 
larger washer being on the side near the handle of the pump. 

Pattern for Reservoir (Fig. 203).— The pattern for the body of 
the reservoir is a rectangle whose length equals (4X7r)+| in. for 
locks, and whose width equals 5 in. A center line is drawn and the 
J^-inch hole for the tube is located upon it. A hole for the f -inch 
screw can top is also located as shown in Fig. 203. The pattern 
for the ends of the reservoir is a circle 4 in. in diameter to which 
is added a |-inch burr to act as a lap for soldering the ends to 
the body. The brace is a rectangular piece of tin f in. wide. The 
length of the brace is taken directly from the profile as it appears in 
Fig. 199. 

Schedule of Materials. — When making a drawing of an article 
that has many parts, a schedule of material is included in the 
drawing. This schedule saves a large amount of description 
regarding material, etc., that would otherwise have to appear on 
the drawing for each part, thereby complicating the drawing and 
making it more difficult to read. 

67. Related Mathematics on Atomizmg Sprayer.— In planning 
an article that is to be manufactured the draftsman must con- 
stantly strive to keep the cost as low as possible. The largest 
items entering into the cost are material and labor. The various 
parts must be so designed that they will "cut to advantage" from 



COMBINATION IS OF VARIOUS SOLIDS 



151 



the stock sizes of sheets. However, there are cases where a small 
amount of extra wastage will be more than compensated for by the 
saving in labor. Figure 204 shows a layout that would preclude 
the possibility of using the squaring shears and the circular shears 
for cutting out the blanks. 

It is evident that cutting must be done with the hand snips. 
While this would be desirable if but one sprayer were to be made, 
it would result in an increased labor cost in quantity production. 



Pomp tSorr«l 

\r 



Reservoi 




raca. I 



Fig. 204. 



Fo 



ur Pump Bon 'els 



[sSdHs: 



r 



EndiY+or RcKcrvoir 



"^ 





Fig. 205. 
Figs. 204-205.— Plan for Cutting Atomizing Spray Pump Parts from Sheets. 



Figure 205 shows the method of arranging the pump barrel and 
brace patterns on the sheet in such a way that they can be cut in 
the squaring shears. The ends for the reservoir and the pump 
nozzles are arranged for cutting in the squaring shears by cutting 
along the dotted hnes. After the sheet is cut into blanks, the cir- 
cular ends may be cut true to shape in the circular shears. The 
nozzle, however, will have to be marked from a master pattern and 
the curved edges cut by the hand snips. 



152 SHEET METAL DRAFTING 

Problem 35 A. — Show by means of sketches how the patterns 
for the atomizing sprayer should be arranged on the sheet in order 
to obtain the greatest saving in material and labor in the manu- 
facture of twelve complete sprayers. 

Problem 35B. — What is the percentage of waste per sprayer? 



COMBINATIONS OF VARIOUS SOLIDS 153 

Problem 36 
ASH PAN WITH SEMICIRCULAR BACK 

68. Ash Pan with Semicircular Back. — Figure 206 shows the 
plan of an ash pan having a semicircular back. The sides and back 
flare 1| in., while the front of the pan flares IJ in. Figure 207 
shows an elevation of the pan. The pan is made from five pieces 
of metal (two sides, front, back, and bottom), all joints being 
double seamed. The top is reinforced with No. 8 wire. 

Pattern for Semicircular Back. — The plan and elevation are 
drawn according to the dimensions given in Figs. 206 and 207. An 
extension hne is carried downward from the center from which the 
semicircular ends are drawn. Another extension line from the 
slant height (line 16-17 of Fig. 207) is drawn to intersect the first 
extension hne, thereby locating the apex of the cone of which the 
semicircular end is a part. With the apex as a center, and a 
radius equal to the distance from the apex to point 17 in Fig. 207, 
the arc of stretchout, Fig. 208, is drawn. The spaces 2, 3, 4, 5, 6, 
7, and 8, taken from similarly numbered spaces in Fig. 206, are 
set off upon this arc. The pattern is completed by an arc drawn 
from point 16, using the apex as a center. One-half inch locks are 
added to both edges of the pattern, a |-inch wire edge to the top 
and a j^-inch single edge to the bottom of the pattern. 

Pattern for Sides of Pan. — Extension lines are dropped from 
points 9, 13, and 8 of Fig. 206. At any convenient location the 
horizontal line 9-8 of Fig. 209 is drawn. Parallel to this line, and 
at a distance equal to line 16-17 of Fig. 207, the horizontal line 
12-13 of Fig. 209 is drawn. The intersections of these hues with 
the extension hues previously drawn will determine the location of 
points 12 and 13. Lines 8-12 and 9-13 are drawn. A |-inch wire 
edge is added to the top, and a j^-inch single edge to the bottom of 
the pattern. A ^-inch double edge is added to side 9-13 for double 
seaming. These patterns must be formed "right and left" in 
making the pan. The rivet holes for the bail ears must be located 
as shown in Fig. 209. 

Pattern for Front of Pan. — The line 9-1 of Fig. 211 is drawn 
equal in length to line 9-1 of Fig. 206. A hne is drawn parallel to 
this hne and at a distance equal to line 14-15 of Fig. 207. Per- 
pendiculars are dropped from points 1 and 9 of Fig. 211 cutting 



154 



SHEET METAL DRAFTING 



the line last drawn. A distance of l| in. is measured in from each 
perpendicular in order to locate points 10 and 13. Lines 1-10 



•Seam 



Wire Edge 



2 Double Edge 




Figs. 206-212. — Ash Pan with Semicircular Back. 



and 9-13 complete the pattern. Three-sixteenths inch edges are 
added to the sides and to the bottom, and a |-inch wire edge to 
the top of the pattern. 



COMBINATIONS OF VARIOUS SOLIDS 155 

Pattern of Bottom of Pan. — The profile of the bottom, Fig. 210, 
of the pan as shown in Fig. 206 should be reproduced, and a 
|-inch double edge added to all sides of this profile in order to 
double seam the bottom of the pan to the sides and ends. 

The Bail. — The bail is made from galvanized j-inch rod, and is 
attached to the pan by bail ears. The bail ears are located ''off 
center" to assure steadiness when carrying the pan. Figure 207 
gives the location of the bail ears. An elevation of the bail, as 
shown by Fig. 212, is drawn. If care is taken to give center line 
measurements, the workman in the shop can "scale" his dimen- 
sions directly and, therefore, a pattern for the blank will not be 
needed . 

69. Related Mathematics on Ash Pan. — Problem 36 A. — What 
is the area of Fig. 208? 

In solving this problem use the formula — - — XlI = ATesi 

A 

B = length of longest arc 
b — length of shortest arc 
i7 = length of "line a" 

Problem 36B. — What is the total area of the sides (two wanted) 

as shown in Fig. 209? 

p I I, 
• Use the formula X ^ = Area 

A 

in which 

B = lower base (over-all dimensions) 
6 = upper base (over-all dimensions) 
H = total height between bases 

Problem 86C.— What is the total area of Fig. 211? 

Problem 36D.— What is the area of Fig. 210? Figure 210 is a 
combination of a rectangle and a semicircle. 

Problem 36E. — How long must the wire be to stiffen the top 
edge of the pan? How much rod is needed to make the bail? 

Problem 36F. — Number 24 gage black steel weighs 1 . 02 lb. per 
square foot. How much will the pan weigh exclusive of the bail 
and the top wire? 



156 SHEET METAL DRAFTING 

Problem 37 
ROTARY ASH SIFTER 

70. The Rotary Ash Sifter. — This problem on the rotary ash 
sifter presents a composite of nearly all of the pattern principles 
that have been given in the preceding problems. 

Figure 213 shows a section and Fig. 214 an end elevation of the 
sifter. Extension lines should be used to establish the elevation 
positions of the views, but these are omitted from the drawing 
to avoid confusion of lines. 

Pattern of Side (Fig. 215). — A line of stretchout is drawn and 
upon it is placed the spacing between letters A, K, B, J, C, F, D, 
and H as shown in Fig. 214. Measuring lines are drawn through 
each of these points. It should be noticed that points E and D, 
G and H, M and A fall on the same horizontal lines of Fig. 214. 
Because of this, they should be similarly placed in the stretchout, 
Fig. 215. Starting from point A of Fig. 213, an extension line 
intersecting measuring line A of Fig. 215 should be dropped. In 
like manner all points of intersection are located. Three-sixteenths 
inch single edges are added, where shown, to provide for double 
seaming, and a f-inch edge which is to be bent at right angles 
to receive the hook of the sliding cover, is allowed for. 

Front End of Hopper (Fig. 216). — A line of stretchout is drawn 
first. Upon this line the distance MK of Fig. 213 is laid off. 
Measuring lines are drawn through these points. Extension lines 
from each end of the lines M and K are dropped until they inter- 
sect the measuring lines of Fig. 216. These points of intersection 
are connected by four straight lines to obtain the outline of the 
pattern. The necessary allowances, as shown, should be added. 
A notch must be cut out to provide for each hinge strap which is 
to be folded over the exposed wire and riveted to the cover. 

Patterns foi- the sliding cover, front end of outlet, and bridge 
are developed by the same method as was the front end of hopper. 
The spaces KJ, JH, and EFG are taken from correspond- 
ingly lettered spaces of Fig. 213. The hook, shown in Fig. 218, is 
made of 1 in. X i in. band iron. 

Pattern for Cover of Barrel (Fig. 220). — The cover is made from 
one piece of metal, the rim being "flanged" as described in Chap- 
ter VII for the treatment of the ash barrel bottom. The width or 



COMBINATIONS OF VARIOUS SOLIDS 



157 



the opening is represented by line DE of Fig. 214, and the length by 
line D-E of Fig. 213. A |-inch double edge is added to the long 
sides and a j^-inch single edge to the short sides of the opening 
The allowance for flanging must be computed by the formula 




Pattern of &arreu Cover 



Figs. 213-221.— Rotary Ash Sifter. 

given in Chapter VII. Outside of this allowance a f-inch wire 
edge is provided for covering the wire. 

Pattern for Hopper Cover (Fig. 221). — This pattern is a rect- 
angle whose length equals line AM, Fig. 214, plus f in. for clearing 



158 



SHEET METAL DRAFTING 



-*iiV- 




Apcx 



End Elevation of Sifter 

/7| Wirg Edge 



Pattern ofGalv. Iron Shield 
Fig. ZU 




< — 1^ 
0- hl^ 



Fig. tZ9 

Pattern of Galv. Screen 



Pattern for Rear End ^'oaUMoop 
OF Sifter ^ ""^ 

Fie. £25 



FlG.t 

5iDE Elevat 




Front Elevation 
Fig. zzi 



Figs. 222-229.— Patterns for Rotary Ash Sifter. 



COMBINATIONS OF VARIOUS SOLIDS 159 

the wire, and whose width equals Une AM of Fig. 213, plus f in. 
for clearance. Rivet holes for the hinge straps should be laid out 
carefully. A double edge is provided on three sides to be formed 
according to the sectional view, A |-inch hem will serve to stiffen 
the cover at the hinged edge. 

Pattern of Rotating Screen (Fig. 222). — The rotating screen is 
made of 3 mesh No. 18 wire galvanized netting. It is in the form 
of a frustum of a cone as shown by Fig. 222. A cast-iron frame, 
shown to the left of Fig. 222, is provided for each end of the screen. 
The outer face of this frame is tinned and the netting is soldered 
to it. The edges of the screen have a f -inch lock turned outward, 
and a galvanized-iron clinch strap is slipped on, hammered down, 
and the whole seam 'Hacked" with solder. The pattern of the 
frustum, Fig. 223, is obtained in the manner described in Chap- 
ter V. 

Pattern of Rear End (Fig. 225). — A rear elevation of the sifter 
is drawn as shown in Fig. 224, all dimensions being taken from 
Figs. 213 and 214. A line of stretchout is drawn and upon it are 
set off the spaces AC and CD of Fig, 213. Measuring lines 
can now be drawn and extension lines dropped from points A, C, 
and D of Fig. 224, Straight lines connecting points of inter- 
section will give the outline of the pattern, A |-inch wire edge 
should be added to the top, and |-inch double edges to the other 
three sides. 

Pattern for Galvanized Screen (Fig. 229). — Figure 226 shows that 
part of the sectional view. Fig. 213, that has to do with the screen 
and shield, while Fig. 227 is a front elevation. The pattern for 
the galvanized iron shield is copied directly from Fig. 227, and the 
necessary laps added as shown by Fig. 228. The line PR is 
extended to the right of Fig. 227, making RW equal in length to 
BS of Fig. 226. With i? as a center, arcs are drawn from points 
15, 5, 6, 7, 8, and N, cutting the line RT, which is drawn at right 
angles to RW. At any convenient point, a line PR, Fig. 229, 
is drawn equal in length to line PR of Fig. 227. With R and P 
of Fig. 229 as centers, and a radius equal to WN of Fig. 227, arcs 
interesting at point N are drawn in. With. P and R, Fig. 229, as 
centers and radii equal to W-8, W-1 , W-Q, TT-S, and TF-15 arcs 
bearing away from point W are drawn. Starting at point N of 
Fig. 229, the distances NS, 8-7, 7-6, etc., should be made equal to 
distances ^"-8, 8-7, 7-6, etc., of the profile of the circle in Fig. 227. 



160 SHEET METAL DRAFTING 

The straight hnes setting forth the flat and curved surfaces should 
be drawn in. A 1-inch lap is added to the curved surfaces. 
71. Related Mathematics on Rotary Ash Sifter. — 

Areas of Trapezoids. 

Problem 37A.— What is the area of Fig. 216? Of Fig. 217? 
Of Fig. 218? Of Fig. 219? Of Fig. 225? 

Area of Circle. 

Problem 37B. — Find the area of Fig. 220, using the over-all 
dimensions. What per cent of the metal is cut away for the open- 
ing? 

Area of Rectangle. 

Problem 37C. — Compute the area of Fig. 221 by using the over- 
all dimensions. 

Areas of Triangles. 

Probleyn 37D. — Divide the pattern of the sides of Fig. 225 into 
triangles and compute the area of each. 

Problem 37E. — What is the combined total area of both sides? 

Problem 37F. — Treating Fig. 229 as a combination of triangles, 
what is its area? 

Frustums of Cones. 

Problem 37 G. — What is the area in square inches of Fig. 223? 

Problem 37H. — By drawing the imaginary line between points 
5 and 13, Fig. 228 would be converted into a trapezoid. What 
would be its area? 

Problem 37-1. — What is the combined area of all of the patterns 
required for the rotary ash sifter? 



CHAPTER XI 
FRUSTUMS OF CONES 



Prob. 
No. 


Job 


DRANA/IM<3 

Objective 


Mathematical 
Objective. 






Co/7e /nfQ/-jectet/ 


Area of frasfu/n 






^y ano-f/f^r 


of co/ie of /»*? 


38 


CUP 

STRAINER. 


co/>e.. 






^^ 


Standord si'z.es 


Yo/i//?7e of frujf- 


39 


TIN DtPPCR 


Pa^er/j of ooss" 


i/m of cone of 
retK, 


40 


LIQUID 
MEASURES 


Peye/o/>/ne/tf /of 

///y /fc/ffern. 
Ifeve/o^/nc/rf of 
'ffa/pd/e 6osB " 


So/t^/'/T^ for 
i//7A/tow/f a^/^ 
mens/o/r. 



Objectives of Problems on Frustums of Cones. 
161 



162 SHEET METAL DRAFTING 

Problem 38 
CUP STRAINER 

72. The Cup Strainer. — This problem introduces the principles 
that apply when two cones intersect each other. Figure 231 is an 
elevation of the cup strainer. The body is a frustum of a cone 
whose apex is noted upon the drawing as Apex No. 1. The handle 
of the strainer is also a cone and miters upon the conical body as 
shown in Fig. 231. When two cones miter upon each other in 
this manner, the miter line must be developed. 

Developing the Miter Line. — The elevation of the cup strainer, 
Fig. 231, should first be drawn according to the dimensions given. 
The body and handle should have their sides extended to deter- 
mine the apex of each. The half-profile of the handle is then drawn 
and divided into equal spaces. After numbering each space, per- 
pendicular hnes are drawn to the base of the cone. From these 
points, extension lines are drawn to Apex No. 2. Directly above 
the elevation, a half plan of the handle is drawn using extension 
Unes to locate the view properly. A quarter-profile, Fig. 230, is 
drawn and the spacing of the half-profile of Fig. 231 is transferred 
in such a way that point 1 falls on the horizontal center line. An 
extension line is drawn from point 2 perpendicular to the base line 
and thence to Apex No. 3. Perpendicular lines are now drawn 
from the points of intersection of extension lines 2 and 3 of Fig. 
231 and the slant height of the body, until they meet the hori- 
zontal center hne of Fig. 230. With one point of the compass on 
the center of the profile of the body, Fig. 230, arcs A and B are 
drawn. Extension lines should now be carried back from the 
points of intersection of arcs A and B with line 2 until they inter- 
sect lines 2 and 3 in Fig. 231. The curved miter hne is now drawn 
in as shown. 

Developing the Pattern for Handle. — From each intersection of 
the miter line in Fig. 231, lines intersecting the slant height should 
be drawn parallel to the base of the cone. These intersections are 
shown by letters, c, d, e, and /, in Fig. 231. The arc of stretchout 
is now drawn and the spacing of the profile transferred with 
numbers to correspond. 

From each point on the arc of stretchout, Fig. 232, lines are 
drawn to Apex No. 2. Arcs should now be drawn from points 



FRUSTUMS OF CONES 



163 



c, d, e, and/ of Fig. 231 over into the stretchout. Points of inter- 
section can be determined by starting from the half -profile, tracing 
the extension line to the miter line, and thence to a correspond- 







Figs. 230-235— Cup Strainer. 

ingly numbered line in the stretchout. A curved line drawn 
through these intersections will give the miter cut of the pattern. 
A |-inch lap should be added to one side of the pattern. 



164 SHEET METAL DRAFTING 

Pattern of Body.— An arc of stretchout, Fig. 234, should be 
drawn whose radius is equal to the distance from Apex No. 1 to 
the top of the body. Six spaces each of which are equal to the 
radius of the top are set off upon this arc. The first and last 
points should be connected to the center from which the arc of 
stretchout was drawn. Another arc is now drawn by using the 
same center and a radius equal to the distance from Apex No. 1 
to the bottom of the body in Fig. 231. A i-inch lap is added to 
one side of the pattern, and a i-inch wire edge to the top edge. 

Figure 233 shows the pattern of the perforated tin strainer. 
The diameter of this blank is equal to the diameter of the body, 
Fig. 231, with a ^-inch lap added all around. 

Figure 235 is the pattern of the rim. Since the rim is a 
cyhnder, its pattern will be a rectangle whose length is equal to 
2| in.Xvr, and whose height is equal to ^ in. plus a i-inch hem. 
A quarter-inch lap should be added to one side of the pattern. 

73. Related Mathematics on Cup Strainer.— Pro&Zem 38 A.— 
What is the area in square inches of the body pattern? 

Problem 38 B.— What is the area of the pattern of the handle? 

Problem 38C. — Show by means of a sketch a method of cutting 
the blanks required for the manufacture of twelve cup strainers, 
that will leave a minimum amount of waste. 

Note.— The formula for the frustum of a cone is given in 
Chapter V. 



FRUSTUMS OF CONES 165 

Problem 39 
SHORT HANDLED DIPPER 

74. Short Handled Dipper. — The dipper presents a problem in 
which three right cones are mitered. The elevation, Fig. 236, is 
drawn and dimensioned. The handle of the dipper is raised 
about 10° above the horizontal, although there is no set rule 
governing this feature. The boss is also drawn to suit the ideas of 
the designer. 

Pattern of Boss. — Figure 237 is a reproduction of the elevation 
of the boss and that part of the dipper that is adjacent to it. The 
sides of the boss are extended to form a right cone. This right 
cone is cut by two planes, the surface of the body being one cutting 
plane, and the hne of junction between the boss and the handle, the 
other. A half-profile is drawn directly upon the base of the cone, 
and divided into equal parts. These parts are numbered and 
extension hnes drawn from each division, perpendicular to the 
base of the cone. From each intersection of the base, hnes are 
drawn to the apex. 

A half plan of the boss is drawn directly above the elevation. 
A quarter-profile is attached to the half plan, with divisions and 
numbers that correspond to the profile in the elevation. 

From each point of intersection on the slant height of the body 
dotted extension hnes are carried up to the horizontal center line of 
the plan. Using the center of the top as a center point, arcs are 
drawn from each intersection of the dotted lines and the horizontal 
center hne. These arcs intersect extension lines drawn from the 
base to the apex in the half plan. Perpendicular hnes are now 
carried back to correspondingly numbered extension lines in the 
elevation. A curved hne passing through the points thus obtained 
wiU be the developed miter line. 

An arc of stretchout is drawn using a radius equal to the 
distance from the apex to point 4. Upon this arc are placed 
twice as many spaces as there are in the half-profile, with numbers 
to correspond. Measuring Hnes are drawn from each of these 
divisions, to the apex. 

From each intersection of both miter lines, extension hnes 
are drawn parallel to the base of the cone until they intersect the 
slant height. From each of these points, extension arcs are drawn 



166 



SHEET METAL DRAFTING 



until they meet correspondingly numbered measuring lines in the 
stretchout. Curved lines drawn through these intersections will 
give the miter cuts of the pattern, A small lap is added to one 
side of the pattern. 

The pattern of the handle, Fig. 239, is obtained in exactly the 




Figs. 236-239.— Short Handled Dipper. 

same way as was the handle for the Cup Strainer, and needs no 
further explanation. 

The pattern for the body is that of a frustum of a right cone. 
This development has been explained in previous problems and 
will not be shown on this drawing. It may, however, be men- 
tioned that the bottom of a dipper is always double seamed 
to the body, and, therefore, proper allowances must be added to 
the pattern for this purpose. 



FRUSTUMS OF CONES 167 

75. Related Mathematics on Short Handled Dipper. — Volume 
of a Frustum of a Cone of Revolution. — The frustum of a cone 
has a circular top and a circular base. These are known as 
the upper and lower bases of the frustum. The altitude of the 
frustum IS the shortest distance between the upper and lower 
bases, and is always measured perpendicularly. The volume of a 
frustum is found by adding together the area of the upper base, the 
area of the lower base, and the square root of the product of the 
upper base area times the lower base area; the sum of these quan- 
tities is then multiplied by one-third of the altitude. 

Expressed as a formula 

in which V=(B+b-\-VBXb)XH ^3 

V = Volume 
B = Upper base area 
b = Lower base area 
^ = Altitude 

In applying this formula to Fig. 236, the areas of the upper and 
lower bases must first be computed. 

Area of Circle = D^X. 7854 (Chapter II) 
6 J 2 X . 7854 = 30 . 68 area of upper base 
4| 2 X . 7854 = 15 . 904 area of lower base 

Known values can now be substituted in this formula 

7= (30. 68+15. 904+ V30. 68X15. 904) X3|-^3 

Performing the arithmetic: 



5 


15.904 
30.68 


V487.9347 22.08+ 


Q3 . q_/^\/^_ii 


4 

42 087 
84 
4408 3 9347 


127232 
95424 

477120 

487.93472 




3 5264 



4083 



168 SHEET METAL DRAFTING 

The formula will now stand, 

F= (30.68+15.904+22.08) Xli 
or 85.83 cu. in. 



Solution ■ 



15.904 68.664 

30.68 1.25 

22.08 

343320 



68.664 137328 

68664 



85.83000 Ans. 85.83 cu. in. 

There are 231 cubic inches in one gallon and the capacity ot this 

85 83 
dipper would be „ or . 37 gallon. 

Problem 39 A. — What would be the capacity in cubic inches of a 
dipper whose dimensions are as follows: Diameter of top, 7j"; 
diameter of bottom, 5j"; altitude or depth, 4^"? 

Problem 39B. — What would be the capacity in quarts of the 
dipper described in Problem 39A? 



FRUSTUMS OF CONES 169 

Problem 40 
LIQUID MEASURES 

76. Liquid Measures. — The body of the measure should first 
be drawn and its side extended to locate the apex. Figure 240 
shows such a view of the body with a half -profile attached to its 
top edge. This half -profile is divided into equal parts, and 
extension lines carried from each diidsion to the top edge of the 
body. 

Pattern of Body. — With a radius equal to the distance from the 
apex to point 7 of Fig. 240, an arc of stretchout, Fig. 241, is drawn. 
The spacing of the half-profile is transferred to the arc of stretch- 
out with numbers to correspond. Straight Hues are now drawn 
from the apex through points 1 and 7, continuing downward indefi- 
nitely. An arc drawn from the apex, with a radius equal to the 
distance from the apex to the point J, completes the half pattern of 
the body. A j-inch wire edge is added to the top of the pattern. 
A j-inch lock and a |-inch single edge for double seaming are added 
as shown in Fig. 241. The half pattern is revolved about fine 7 
of Fig. 241 in order to obtain the full pattern. 

Elevation of Lip. — The Hp should now be added to the eleva- 
tion of the body. In constructing the elevation of the Hp, a point 
H is selected 1| in. below the top edge of the body on the center 
line. Straight fines are drawn from this point through and beyond 
points 1 and 7. Point A is located | in. from point 1, and point G 
is located 1| in. from point 7. The fine AG completes the eleva- 
tion. Extension lines must now be drawn from the apex (point H) 
cutting the top of the fip at points A, B,C, D, E, F, and G. From 
these points, horizontal fines are now shown intersecting the fine 
HG, which is the slant height of the cone. 

Pattern of Lip, — In any convenient space, an arc of stretchout. 
Fig. 242, is drawn with a radius equal to line ^-1 of Fig. 241. 
Twice as many equal spaces are set off on this arc as there are 
spaces in the half -profile, with numbers to correspond. Measur- 
ing fines are now drawn from the point H through and beyond 
each division of the arc of stretchout. The distance l-A, on fine 
HA of Fig. 242, is exactly equal to distance l-A of Fig. 240. 
Distances HB, HC, HD, HE, and HF of Fig. 242 are ob- 
tained by measuring the distances from point H to the inter- 



170 



SHEET METAL DRAFTING 



sections of the horizontal lines that were previously drawn from 
points B, (7, D, E, and F, and the slant height (line HG) of Fig. 



i Wire Edge 




Figs. 240-244. — Measure for Liquids. 



240, Distance HG is taken directly from hne HG of Fig. 240 
because it is the slant height, and, therefore, a true length. 



FRUSTUMS OF CONES 



171 



A curved line traced through the points thus obtained completes 
the pattern for the hp. A J-inch wire edge and a j-inch lap 
should be added as shown. 

Pattern of Boss. — The pattern of the boss, Fig. 244, is obtained 
by drawing a profile in its proper location as shown. This profile 
is divided into four equal spaces and extension hnes carried from 
each division, intersecting the elevation of the handle. A line of 
stretchout is next drawn with spacing and nmnbers to correspond 
to the profile. Measuring lines should now be drawn and inter- 
sected by extension lines drawn from the elevation. A curved 
line drawn through the points thus obtained will give the pattern 
of the boss. 

Pattern of Handle (Fig. 243). — Upon any straight line, the 
distance around the handle profile, shown in Fig. 240, is set off. 
Perpendiculars are erected at the first and last points. The upper 
end of the handle is obtained by setting off j^ in. on each side of 
the horizontal line, and the lower end by setting off y^ in. on each 
side of the horizontal hne. The pattern is completed by adding 
j^-inch wire edges to each side of the handle. The pattern for the 
bottom (not shown) is a circle whose diameter is equal to the 
finished diameter (4| in.) plus a j-inch edge for double seaming 
to the body. The actual diameter of the pattern would be 
4| in. + i in. + l in. =4f in. 

77. Related Mathematics on Liquid Measures. — Problem 
IfiA. — Compute the capacities of the measures given in the follow- 
ing table: 





Standard Sizes 


FOR Flaring Liquid Measures 


Diameter of 


Diamet 


er of Height 


Capacity. 


Bottom. 


Top 


(Altitude). 


2i in. 


233^ i 


n. 2 in. 




2| ' 




2i ' 


3J " 




3 ' 




21i' 


4^" 




4 ' 




3i ' 


4f " 




5A' 




31 ' 


7A" 




61 ' 




5 ' 


8A" 




Sf ' 




61 ' 


9f " 




101 ' 




8 ' 


lOi " 




11 ' 




• ' 81 ' 


12A" 




vi\ ' 




9i ' 


12A" 





172 SHEET METAL DRAFTING 

Problem JfOB. — A customer wishes a 2| gallon flaring measure. 
The bottom of this measure is to be 9" in diameter, and the top 
1" in diameter. How high must the measure be made to fulfill 
these requirements? 

Example of Problem 40B. — Suppose the dimensions were 6" 
diameter of bottom, 4" diameter of top, and the measure was to 
hold 3 quarts. 

Original Formula F= (B-\-b-\-VBXb) H^3 

Transposing, H^3 =-V-^(B-\-b-}-VBXb) 

In this formula, H = Altitude, which is the unknown 

F = Volume = 3 quarts or 173.25 cu. in. 

B = area of lower base = 6^ X . 7854 = 28 . 274 sq. in. 

b =area of upper base =4-X .7854== 12.566 sq. in. 

Substituting known values in above formula, 

^^3 = 173. 25 -^ (28. 274+ 12. 566+ V28. 274X12. 566) 
and iy^3 = 173.25-^59.69 
H-^3 = 2.9 

^ = 2. 9X3 = 8. 7 in. Ans. S. 7 in. high. 



CHAPTER XII 
RETURN AND FACE MITERS 



Prob. 
No. 



Job 



Drawi im g 
Objective 



Mathematical 
Objective 



41 




Return Miter 



pr//7C/p/es of<^eye/- 

Ofme/ff- used for 

my^fj iff /7?0yW/JfS 
and cor/? ices. 



AZ 




p/es of /77/yer deve/- 
0/>/nent to square 
re fur/? /n/ten. 



£'sf/ft7at/'/i^ vve/fM 
and CO if ofcon- 
dacfor J?ead. 



COMOUCTOR HEAD 



43 




/^pp/^/'/jp ■ffrepr/n- 
c/'p/es of rr?/fer 
devefop/ne/rf fo 
Sfuare facQ /niffrs 



FACE Miter 



44 




Window Cap 



App/y//T^ ff?e pr//7- 
c/p/es to fac e. 
/fiifers at ofAer 
^an r/'gtif a/r^fes 



Objectives of Problems on Return and Face Miters. 
173 



174 SHEET METAL DRAFTING 

Problem 41 
SQUARE RETURN MITER 

78. Square Return Miter. — Figure 245 shows the profile of a 
moulding. Mouldings are seldom of standard design, although the 
architect builds up a given design from standardized parts or 
members. In the profile shown, the compound curve is known as 
an ogee. This shape is encountered more frequently in mould- 
ings than any of the others. Line 9-10 of Fig. 245 is often referred 
to as a "fascia," which is a plain band or surface below a moulding. 
Line 10-11 of Fig. 245 forms the drip of the moulding since it com- 
pels the water, flowing down the surface of the moulding, to drop 
off. The hues 11-12 and 12-13 are called fillets. Fillets are 
narrow plain surfaces used to separate curved members of a 
moulding, or to finish a moulding. On a moulding of this design 
the fillet 12-13 is intended to enter a reglet (slot) in the side wall of 
the building. 

In drawing the ogee curve, a square, 3-J.-9-5, Fig. 245, is 
drawn whose sides are equal in length to the desired height of the 
member. A horizontal center Une CD is then drawn and from 
points C and D the curves forming the ogee are drawn. This 
gives an ogee whose height equals its projection. Architects 
often modify this curve in order to gain height without attaining 
too great projection. 

After the profile is drawn it should have all of its curved lines 
divided into equal spaces. Numbers should be placed at each 
angular bend (vertex) and at each division of the curved fines. A 
line dropped from points 2 and 13 will show the entire width or pro- 
jection of the moulding, Fig. 246. If this width is carried around 
the corner at an angle of 90°, a plan of a square return miter will 
result. The miter line, as shown in Fig. 246, must always bisect 
the angle formed by the sides of the moulding. 

Lines should be dropped from every point in the profile, down- 
ward through the plan, an indefinite distance. A line of stretch- 
out should now be drawn at right angles to the side of the plan as 
shown in Fig. 247. Every space in the profile is now transferred 
to the line of stretchout, care being taken to get them in theu* 
proper sequence, and to have the numbers correspond. Measur- 
ing lines are drawn through each division at right angles to the line 
of stretchout. 



RETURN AND FACE MITERS 



175 




Figs. 245-249.— Square Return Miter. 



176 SHEET METAL DRAFTING 

Starting at point 1 of the profile, the extension Hne should 
be followed downward until it intersects line 1 of the stretchout. 
In hke manner all of the intersections should be located and marked 
with small circles. The miter cut may now be drawn in by con- 
necting the intersections of the stretchout by straight and curved 
lines. It should be observed that curved lines in a profile will 
always produce curved lines in the pattern, and straight lines in 
the profile will produce straight lines in the pattern. 

Figure 248 represents the moulding carried around another 
corner. Extension lines are carried upwards from this view and 
are intersected by correspondingly numbered extension lines from 
the profile. In this manner an elevation, Fig. 249, of any miter 
may be projected. 

It should be observed that the plan. Fig. 246, plays no part in 
the development of the pattern, the extension lines from the profile 
remaining unchanged in passing through this view. This is true 
of all square (90°) return miters. However, if the miter was at 
any other angle, say 87°, the extension lines would be deflected by 
the changed position of the miter line, and a plan view would be 
absolutely necessary for the development of the pattern. 



RETURN AND FACE MITERS 177 

Problem 42 
CONDUCTOR HEAD 

79. Conductor Head. — Conductor heads are used to ornament 
the conductor pipes of a building and are usually placed at the 
point where the "goose neck" from the gutter enters the conductor. 
As shown by the dotted lines in Fig. 250, a short piece of rectangu- 
lar or round pipe is carried through the head in order to give a more 
direct travel to the water. 

Conductor heads are made in a great variety of shapes and 
sizes. On the better class of buildings, they are designed to har- 
monize with the particular style of architecture adopted. 

Figure 250 shows a front elevation of a conductor head. Since, 
as was explained in the preceding problem, the pattern may be 
taken directly from the profile as it appears in the elevation, the 
curved lines may be divided into small parts. It will be noticed 
that the space between points 20 and 21 is less than that between 
the other points. This is perfectly permissible, as long as the same 
distance appears between points 20 and 21 on the line of stretchout, 
and saves much time that would otherwise have to be spent in 
making all spaces exactly equal. The dividers are set at any 
radius not too large and the curve is spaced off, allowing the last 
space to come wherever it may. 

A center line is drawn in Fig. 250 and extended downward to 
serve as a hne of stretchout for Fig. 252. The spacing of the pro- 
file is now transferred to this line and numbered to correspond. 
Measuring lines are drawn at right angles to the line of stretchout 
and intersected by extension lines dropped from correspondingly 
numbered points in Fig. 250. The miter cut of one side of the 
pattern is now drawn in. 

Since both sides of the front (as divided by the center line) are 
symmetrical, the distances from the center line of Fig. 252 to each 
point in the miter cut should be transferred to the other side, there- 
by obtaining the necessary points for drawing in the other miter 
cut of the pattern for the front. 

The side elevation. Fig. 251, is now drawn and the pattern de- 
veloped by the method already described for obtaining the pattern 
of the front. Laps are added and notched as shown in Fig. 253. 
Two of these patterns must be cut from the metal and while they 



178 



SHEET METAL DRAFTING 



are both alike, it is evident that they must be formed in pairs; 
that is, one right-hand and one left-hand, in order to attach them 
to the front of the head. 

The pattern of the back, Fig. 254, is obtained by reproducing 




PATTERN OF 

FRONT Fro. Z5t 



PATTERN OF 
SIDES Fie. £53 



PATTERN OF 

BACK Fl6. ZSA 



Figs. 250-254.— Conductor Head. 

the outline of the front elevation, Fig. 250, and adding laps, which 
are notched as shown. 

80. Related Mathematics on Conductor Head. — Since all 
conductor heads are composites of the surfaces of many different 
solids, it is unpracticable to attempt to compute their exact sur- 
face area. The most accurate method of obtaining the cost of 
material entering into their manufacture is to lay out a full size 



RETURN AND FACE MITERS 179 

pattern, arrange the several pieces so as to produce a minimum of 
waste, and compute the area of the rectangle thus obtained. 

Problem 4^ A. — Show how you would arrange the blanks for the 
conductor head in order to produce as little waste as possible. 

Problem 4-^B. — What is the area in square feet of the metal 
required as shown by Problem 42A? 

Problem 4^0. — What would be the weight of the metal if it were 
16 oz. copper? 

Problem 42D. — Allowing three hours' labor at $1.35, and 
30 cents for solder, what would be the selling price of the head 
if 30 per cent of the cost price were added for profit? 



180 SHEET METAL DRAFTING 

Problem 43 
FACE MITERS 

81. Face Miters. — A face miter may always be distinguished 
from a return miter by the fact that the miter Hne can be seen in 
the elevation; whereas, the miter hne of a return miter always 
appears in the plan. 

Figure 255 gives the same profile as was used in Fig. 245. This 
profile is used again in order to afford the student an opportunity 
to compare the two types of miters and note wherein the difference 
lies. 

The ogee is divided into small spaces and numbers placed at 
each point of the entire profile. 

Extension fines are carried over to the right from points 1 and 
11 to form the outhne of one leg of the miter. These fines are then 
intersected by a miter line drawn at an angle of 45° since the miter 
itself is a square, or 90° face miter. The other leg of the miter is 
now drawn and lines added to complete the elevation as shown by 
Fig. 256. 

A line of stretchout, Fig. 257, is next drawn and the distances 
between points of the profile transferred in their proper sequence, 
with numbers to correspond. Measuring lines are drawn through 
each of these points at right angles to the line of stretchout. Start- 
ing from point 1 of the profile an extension line is carried to measur- 
ing hne 1 of the stretchout. In like manner all other points of 
intersection are located in the stretchout. Any tendency to "short- 
circuit" this operation should be guarded against by referring 
back each time to the starting point in the profile. By neglecting 
to do this, mistakes are apt to occur, which can be detected only 
after the parts are formed up and the assembling process has 
begun. The miter cut of the pattern should now be drawn in, and 
dots placed on the lines that are to be bent in the cornice brake. 

As was the case with the square return miter, the pattern can 
be taken directly from the profile. The main consideration is the 
proper placing of the profile with reference to the direction in which 
the extension fines from the profile are to be drawn. A mistake of 
this nature will result in a face miter when a return miter was in- 
tended; or, a return miter when a face miter was desired. Also, 
as was the case of the return miter, the extension lines can be taken 



RETURN AND FACE MITERS 



181 



from the profile, only in case the miter is at an angle of 90°. Other- 
wise an elevation must be drawn, as the miter line will deflect the 




(^ 



f^ 



extension lines as they pass through the elevation in a miter other 
than one of 90°. 



182 SHEET METAL DRAFTING 

Problem 44 
WINDOW CAP 

82. Window Cap.— Figures 258 and 259 show the elevation and 
profile of a window cap in the form of an angular pediment. It is a 
combination of two horizontal mouldings having square return 
miters on their outer ends and two inchned or rake mouldings that 
miter upon each other at the center hne, and with the horizontal 
mouldings at their lower ends. The triangular space beneath the 
rake mouldings contains a sunken panel. 

This problem presents two new features; namely, a face miter 
at other than right angles, and a sunken panel. A profile is 
"drawn in" one of the rake mouldings in order to show the amount 
of "sink," and the method of joining the panel to the mouldings. 

The details for a job of this nature are always furnished by 
the architect. The exact measurements must be taken at the 
building, where it is often found that a given set of windows will 
vary from | in. to \ in. in width. This variation is taken care of by 
lengthening or shortening the horizontal mouldings. This can be 
done by the cutter since it does not affect the miter cuts. 

The elevation, Fig. 258, must be carefully drawn, care being 
taken to draw the rake mouldings at their proper angles. The 
miter Hues must also exactly bisect the angle of the miter. 

The profile. Fig. 259, is next drawn and the curved Hne divided 
into small spaces. Each point and vertex are then numbered. An 
extension Hne is now carried from the point of intersection of the 
miter Hne and the sunken panel, point A of Fig. 258, over into the 
profile as shown by point A of Fig. 259. 

From each point in the profile, extension Hues are carried over 
into the elevation, Fig. 258, until they intersect the first miter 
Hne. These extension lines are now carried parallel to the outline 
of the rake moulding until they intersect the second miter line, 
which is also the vertical center line of the entire elevation. 

A Hne of stretchout. Fig. 260, is now drawn at right angles to the 
side of the rake moulding. Upon this Hne the exact spacing of the 
profile from points 1 to A inclusive should be laid down. Then by 
referring to the profile that is drawn in the right-hand rake mould- 
ing, it win be seen that distances 18-^, AB, and BC must be 
added to the line of stretchout beyond point 18. Extension lines 
should now be drawn at right angles to the sides of the rake moulding 



RETURN AND FACE MITERS 



183 



from each intersection of both miter lines of the rake moulding. 
Points of intersection in the stretchout may be determined by- 
following each extension line from its source in the profile to a 
correspondingly numbered line in the stretchout. The miter 
cuts of the pattern are drawn by connecting the points of inter- 
section. As in the case of the side patterns for the conductor head, 




Figs. 258-263.— Window Cap. 

one pattern will suffice for both rake moulds, but they must be 
formed in pairs for assembling. 

Another line of stretchout is now drawn at right angles to 
the base line of the horizontal moulding- as shown in Fig. 261. 
Upon this Une should be laid down the spacing of the entire profile 
including the point A . Measuring fines are drawn through each 
point at right angles to the line of stretchout. 



184 SHEET METAL DRAFTING 

The measuring lines should now be intersected by extension 
hnes dropped from the miter line between the horizontal and rake 
mouldings, and from each intersection of the profile which appears 
on the left-hand end of the horizontal moulding (the return miter) . 
Intersections of the stretchout may now be definitely determined 
by tracing each extension line from its source in the profile to a 
correspondingly numbered line in the stretchout, Fig. 261. The 
miter cuts are now drawn. Two blanks from this pattern are 
needed and they must be formed in pairs. 

Using the same set of measuring hnes, extension lines are now 
dropped from each point in the profile, Fig. 259, until they inter- 
sect correspondingly numbered lines. Lines connecting these 
points will give the outline of the pattern for the ends, Fig. 262. 
A lap is added to the top of the pattern for joining to the ''wash" 
(top surface) of the horizontal moulding. From this pattern two 
blanks, formed in pairs, must bo cut. 

The pattern for the panel is obtained by reproducing the sur- 
face A,E, F,G, H of Fig. 258. The edge (fine BC of profile in 
Fig. 258) for joining the panel to the rake moulding must be added 
to four sides of this surface. The spaces between points 19, 20, 
and 21 of the profile in Fig. 258 must be added to the base of 
this surface as in Fig. 263. It should be noticed that the space 
19-20 of this profile is less than space 19-20 of Fig. 259, because of 
the sunken panel. From this pattern but one blank is cut, which 
is formed according to the profile in Fig. 258. The sunken panel 
leaves small openings along the Hnes AE and FG of Fig. 258, 
which may be closed by allowing a "tab" at Hne ^-19 of Fig. 261. 
However, as this wastes material a small piece of metal may be 
cut to shape, and inserted after the window cap has been assembled. 

The distinction between a rake moulding and a raked or raking 
moulding should be noted: 

A rake moulding is simply a moulding that is inclined to the 
horizontal. It has the same profile as the horizontal moulding to 
which it is joined. 

A raked or raking moulding is an inclined moulding that joins 
a horizontal or other moulding that does not he in the same plane. 
It does not have the same profile as does the moulding to which it 
is joined. It takes its name from the fact that its profile must be 
altered or raked in order to join with the other moulding, 



CHAPTER XIII 
TRIANGULATION OF SCALENE CONES 



Prob. 

No. 



Job 



Drawing 
Objective 



Mathematical 
Objective 



45 




pr/hap/es i//?(^er- 
/y//?^i(/?e sf(/dy 
of rr/a/?^(//af/off. 



Scalene 

CONIE 



46 




^pp/y//?f xa/e/?e 
co/?e/o cA/ff?/7ey 
top pater/?. 
One prof //e (fiif/ded. 



SQUARE TO ROUND 
TRANSITION 



OVAL TO ROUND 
TRANSJTi0^4 



47 




f/t//?^. 

6 of A prof/fes 



Co/7?jc>t/f//?g 
e(ft//y<7/e/?f (preas. 
Area o/oyaA 



Objectives of Problems on Triangulation of Scalene Cones. 
185 



186 



SHEET METAL DRAFTING 



Problem 45 
SCALENE CONE 

83. The Scalene Cone.— Figure 264 shows the top view of a 
scalene cone whose base is represented by a circle and whose apex 
falls in a hne perpendicular to the plane of the base at the point 2. 





\ 






CS) 




v 


Z 




<* 


N. 


o 




<o 


>v 


h 




tJ 


N. 


5 




(J o 


> 


V kJ 
















N. U 










\ 



Figure 265 is an elevation of the cone, showing the apex, point 3, 
on the perpendicular Une drawn from point 2. The hne 2-3 is the 
altitude of the cone. 



TRIANGULATION OF SCALENE CONES 187 

The top view is drawn and the circumference of the circle 
divided into eight equal parts. Straight lines are then drawn 
connecting point 2 with points A, B, C, and D. These are 
known as base Unes, since they are equal in length to similarly 
drawn Hnes on the model; that is, they are true lengths. 

Four horizontal lines, Fig. 266, are now drawn equal in length 
to lines A-2, B-2, C-2, and D-2 of Fig. 264. Corresponding letters 
and numbers are placed at the extremities of these lines. Perpen- 
dicular lines are erected at point 2 of each line, equal in length to 
line 2-3 of Fig. 265. Points A-3, B-3, C-3, and D-d may now be 
connected by straight lines, thereby forming four right triangles. 
The hypotenuses of these triangles are elements of the surface of 
the cone; that is, they are equal in length to lines similarly drawn 
on the surface of the model. That branch of pattern drafting 
known as triangulation takes its name from the fact that the sur- 
faces are developed from a series of triangles whose hypotenuses 
are equal to certain elements — straight lines drawn on the surface 
of the cone. 

A vertical line, Fig. 267, is now drawn equal in length to the 
altitude, line 2-3, of Fig. 265. With point 3 as a center and radii 
equal to hypotenuse S-D,3-C, 3-B, and 3-^, arcs are drawn to the 
left of Kne 2-3. Point D is located by an arc drawn from point 2 
whose radius is equal to distance 2-D of Fig. 266. In like manner 
points C, B, and A are located by arcs drawn from points D, C, 
and B respectively. All of these intersecting arcs have the same 
radii, since the base of the cone was equally divided. A straight 
line 3-A and a curved line A, B, C, D, 2 completes the half pat- 
tern, which may now be copied on the other side of line 2-3 to 
obtain the full pattern. 

It is advisable to make a model by cutting out the triangles 
of Fig. 266, attaching them to the base lines of Fig. 264, and slip- 
ping the envelope, Fig. 267, over this framework. 



188 SHEET METAL DRAFTING 

Problem 46 
SQUARE TO ROUND TRANSITION 

84. Square to Round Transition. — The sheet metal worker is 
often called upon to make square to round transitions. In heating 
and ventilation, square and rectangular pipes are changed to round 
pipes, and ventilators with round shafts are mounted on rectangu- 
lar bases. Wherever the cross-section of a pipe is changed to 
another shape the transformation should be gradual in order to 
avoid excessive friction. 

Figure 268 is a pictorial view of a square to round transition. 
The transition may be considered as being made up of a rectangular 
prism, having a portion of a scalene cone at each corner, the spaces 
between these being filled by triangular-faced pieces. 

Figure 269 shows one-quarter of the transition removed and the 
triangles that are to be used in the development of the pattern 
drawn in their respective positions. 

The Plan. — The plan, Fig. 271, is the first view to be drawn. 
The plan may be divided into four equal parts. It is necessary to 
treat but one part. The center points of two sides of the square 
are first determined as shown by points 1 and 3. That part of the 
circumference between the horizontal and vertical diameters is now 
divided into four equal parts as shown by points A, B, C, and D. 

The base lines are now drawn in, but before drawing them the 
draftsman must determine the order in which he intends to develop 
the pattern. It will simplify the study of triangulation if a stand- 
ard method of development is adopted. Every line should be 
considered as running in but one direction; for instance, the line 
AB should be considered as running from point A to point B 
and not from point B to point A. Furthermore, this line should 
always be read as A to B, and not simply AB. By pursuing this 
method the draftsman is enabled to leave his drawing at any time 
and pick up the "thread" where he left off, upon his return. The 
letters should be confined to one base and the figures to the other. 
Thus in Fig. 271, the order would be 1-A, 2- A, 2-B, 2-C, 2-D, 
and Z)-3. 

The elevation, Fig. 270, may now be drawn, but since the only 
added information it contains is the altitude of the triangles the 
experienced draftsman rarely draws this view. 



TRIANGULATION OF SCALENE CONES 



189 



The Diagram of Triangles. — A series of short horizontal lines, 
Fig. 272, are drawn equal in length and numbered to correspond to 




c3 



pt( 



the base lines of Fig. 271. Perpendiculars are erected at points 2 
and 3, and the several hypotenuses are drawn in as shown. 

The Pattern. — A horizontal straight Une is drawn equal in 



190 SHEET METAL DRAFTING 

length to line 1-2 of Fig. 271. With point 1 as a center and a 
radius equal to the hypotenuse of triangle 1-A, an arc is drawn 
below line 1-2. This is intersected by an arc drawn from point 2 
with a radius equal to the hypotenuse of triangle 2-A. The inter- 
section locates the point A on the pattern. With point 2 as a 
center and a radius equal to the hypotenuse of triangle 2-B, an 
arc is drawn bearing away from point A. This arc is intersected 
by another drawn from point A, whose radius is equal to line AB 
of Fig. 271. This intersection is lettered B. In like manner, 
points C and D are located. Then with point D as a center and a 
radius equal to the hypotenuse of triangle D-3, an arc is drawn 
bearing away from point 2. This arc is intersected by another 
drawn from point 2, whose radius is equal to line 2-3 of Fig. 271. 
Straight and curved lines connecting these points give the outhne 
of the quarter pattern. This is now duplicated on the other side 
of Une 1-A to obtain the half pattern. 

Half-inch locks are added to each side of the pattern, but the 
workman, in forming the locks, should turn but j^ in. It is advis- 
able to construct a model from the plan and diagram of triangles, 
in order to aid in the visualization of the project. 



TRI ANGULATION OF SCALENE CONES 191 

Problem 47 
OVAL TO ROUND TRANSITION 

85. Oval to Round Transition. — The oval to round transition 
is extensively used in hot air furnace heating. In Fig. 275 the oval 
and the circle have the same center, but often the job demands 
that the center of the circle be placed to one side of the oval. How- 
ever, the method of developing the pattern is the same in all cases, 
as long as the planes of the top and the bottom are parallel. 

The Plan (Fig. 275). — The profiles of the upper and lower 
bases should be drawn in their proper positions with a horizontal 
center line for each. Since these profiles have the same center, the 
line AJ divides the figure into two equal parts and, therefore, 
but one-half need be treated. 

Both half-profiles are now divided into equal spaces and each 
division numbered or lettered as shown. The order of develop- 
ment, as explained in Problem 46, should now be determined. 

The Diagram of Triangles (Fig. 276). — Having determined the 
order in which the base lines are to be taken from Fig. 275, short 
horizontal lines equal to each base line are drawn. These are 
shown in Fig. 276, and the order should be carefully studied. Per- 
pendicular lines equal in length to the altitude of the fitting, as 
shown in Fig. 276, are erected at one end of each of these lines. 
The hypotenuses of the several triangles are then drawn in. 

The Pattern (Fig. 277). — A distance equal to the hypotenuse of 
triangle AD is set off upon any vertical line. These points are 
lettered A and 1. With point 1 as a center and a radius equal to 
the hypotenuse of triangle 1-B, an arc is drawn bearing away from 
point A. This is intersected by an arc drawn from point A, whose 
radius is equal to line AB oi Fig. 275, thereby locating point B. 

With B as a center and a radius equal to the hypotenuse of tri- 
angle B-2, an arc is drawn bearing away from point 1. This is 
intersected by an arc drawn from point 1, whose radius is equal to 
the line 1-2 of Fig. 275, thereby locating point 2. 

In Hke manner all points of the pattern may be located. Atten- 
tion is called to the space between letters E and F of Fig. 275. 
Since this is a straight fine it is not divided and, therefore, the space 
EF is greater than any of the others. 

Curved lines passing through the points thus obtained give the 



192 



SHEET METAL DRAFTING 




H 
o 



Q 



2 



TRI ANGULATION OF SCALENE CONES 193 

outline of one-half of the pattern. This is copied on the other side 
of Uhe 1-A in order to produce the whole pattern. 

86. Related Mathematics on Oval to Round Transition. — 

Equivalent Areas. — When two dissimilar profiles contain the same 
number of square inches of surface area, they are said to have 
equivalent areas. When any change of profile occurs in a system of 
pi]3ing, the areas must be equivalent. 

Area of an Oval. — An oval is a rectangle having semicircular 
ends; therefore, its area is equal to the area of some rectangle, plus 
the area of some circle. 

In any oval the diameter of this circle is equal to the width of 
the oval. The rectangle has for its dimensions the width of the 
oval, and the difference between the width and the length of the 
oval. 

Example. — What is the area in square inches of an oval profile 
4" wide and 14" long? 

Diameter of circle = 4" Dimension of rectangle 4" X (14"— 4") 
Area of circle = 4^ X . 7854 . 7854 

16 



47124 

7854 



12.5664 sq. in. ' 

Area of rectangle = 4X10 = 40 sq. in. 

Combined areas =40+12.57 = 52.57 sq. in. Ans. 

Problem JflA. — What are the areas of the following sizes of 



oval profiles? 






(a) 3i"Xl5" 




(6) 4i"Xl4r 




(c) 3" Xll" 




id) 3r'xi5r 




(e) 6" Xl3f" 



Problem 47B. — An 8" round pipe is to be "ovaled down" 
to a width of 3|". What must be the length of the oval? 

(Hint: Subtract the area of a 3|" circle from the area of the 
8'' circle and divide the remainder by the width of the oval.) 



194 SHEET METAL DRAFTING 

Problem 47C. — A furnace man finds the following sizes of oval 
risers have been installed in the partitions: One 3^"Xl3f"; 
one 6"X9i"; one 3^"Xl5i"; and one 3|"Xl4i". What are the 
equivalent round pipe areas for cellar mains to supply each of these 
risers? What is the nearest diameters of the cellar mains if the 
diameters increase by half inches? 



CHAPTER XIV 
TRIANGULATION OF TRANSITION PIECES 



Prob. 
No. 



Job 



Drawing I Mathematical 



Objective 



Objective 



48 




//?c//)pee/ /f/affe . 

secf/M o/JCi/ff//?^ 
p/a/7e. 



Square to Round 



49 




Designing i^e ffff//i^ 
Oua/ fo Round 

I^ary//?^ aJf/fudea. 



Oval to Round 



±. 



Objectives of Problems on Triangulation of Transition Pieces. 

195 



196 SHEET -METAL DRAFTING 



Problem 48 

TRANSITION BETWEEN A SQUARE PIPE AND THE SECOND 
PIECE OF AN ELBOW 

87. Transition between a Square Pipe and the Second Piece 
of an Elbow. — In order to save height the sheet metal draftsman 
is often compelled to design a transition between a square or a 
rectangular pipe and the second piece of a round pipe elbow. In 
other words, the transition takes the place of the first piece of the 
elbow. 

In order to accomplish this purpose it is necessary to incline 
the plane of the top to that of the base of the transition. 

Figure 278 shows the elevation of the small end of a two-piece 
45° elbow with the transition attached in its proper position. In 
drawing this view care must be taken to get the true miter hne 
according to the rules laid down in Chapter III. 

Directly above the small end of the elbow a half -profile is drawn 
and divided into eight equal parts. The divisions are numbered 
as shown, and extension lines carried down through the elevation 
until they meet the miter hne. Numbers are placed on the miter 
line to correspond to the numbering of the half-profile. 

As has already been pointed out in previous chapters, whenever 
a cyhnder is cut by an inclined plane, the section on that cutting 
plane is an ellipse. In order to obtain the proper spacing on the 
pattern a true section on the miter line must be developed in the 
following manner. Line 1-9 of Fig. 279 is an extension of line 
1-9 of the half-profile. Upon this hne should be placed the exact 
spacing of the miter line, and perpendiculars erected at each point. 
These perpendiculars are intersected by extension lines brought 
over from correspondingly numbered points in the half-profile. 
A curved line traced through the intersections thus obtained gives 
a true section on line 1-9 (miter line) of the elevation. 

From each intersection of the miter line of Fig. 278 extension 
lines are dropped vertically for an indefinite distance. The profile 
of the square base of the transition is next drawn in its proper posi- 
tion as shown by A, B, C, D of Fig. 280. 

The horizontal center line, EF, locates the center of the circle 
which is also the profile of the round pipe. The extension lines 
from the miter line of Fig. 278 should divide this circle into equal 



TRIANGULATION OF TRANSITION PIECES 197 

spaces. If they fail to do so, an error in drawing has been made 
which should be corrected before proceeding further. 

The order of development must now be decided upon. The 




Ph 






Ph 



triangles naturally divide into two groups, Group A being those 
triangles having theii' base lines starting from point A, and Group 
B starting from point B. Besides these triangles there is a start- 



198 SHEET METAL DRAFTING 

ing line which is the hypotenuse of a right triangle upon the base 
line E-1 and a finishing Hne upon base line 9~F. 

The diagrams of triangles are now constructed by drawing short 
horizontal lines equal in length to their respective base lines with 
corresponding numbers and letters. Perpendiculars are erected 
at one end of each of these horizontals. Since the plane of the top 
of the transition is inclined, the altitudes of these triangles vary. 
This variation is shown in Fig. 278 where the altitudes of 
various triangles are plainly marked. In determining the altitude 
of any point it should be remembered that the altitude is always 
the perpendicular distance between the plane of the base and the 
point in question. These altitudes should be placed on the proper 
perpendiculars, and in this connection it may be noted that the 
altitude always changes with the number; that is, wherever the 
number 2 occurs the altitude of 2 as shown in Fig. 278 must be 
used. The hypotenuses of the several triangles are now drawn. 

The pattern development is started by drawing a horizontal line 
equal in length to the side DA of Fig. 280. Upon this hne the 
center point E should be placed. A perpendicular line is erected 
at point E equal in length to the hypotenuse of triangle E-1. 
This establishes point 1, and the distance from point A to point 1 
should correspond exactly in length to the hypotenuse of triangle 
A-1. 

Since the center line EF of Fig. 280 divides the figure into two 
equal parts the pattern can be developed on each side of line E-1 
of Fig. 283 simultaneously. The experienced draftsman always 
takes advantage of this fact when a whole pattern is to be devel- 
oped. The line D-1 in Fig. 283 is next drawn, and when a distance 
is laid off from point A a like distance is also laid off from point D. 

With point A as a center and a radius equal to the hypotenuse 
of triangle A-2 an arc is drawn bearing away from point 1. This 
is intersected by an arc drawn from point 1 with a radius equal to 
the distance 1-2 of Fig. 279. In hke manner points 3, 4, and 5 
are established, but it must be remembered that the distances 
between figures must be taken each time from the true section, 
Fig. 279. With point 5 as a center and a radius equal to the hypot- 
enuse of triangle 5-B, an arc is drawn bearing away from point A. 
This is intersected by an arc drawn from point A with a radius 
equal to side A-B of Fig. 280. This establishes point B. 

From B as a center and with the several hypotenuses of Group 



TRIANGULATION OF TRANSITION PIECES 199 

B triangles as radii, points 6, 7, 8, and 9 are established in the same 
manner as were points 2, 3, 4, and 5. 

With point 9 as a center and a radius equal to the hypotenuse 
of triangle 9 to F, an arc is drawn bearing away from point B. 
This is intersected by an arc drawn from point B with a radius equal 
to Hne BF of Fig. 280. This estabhshes point F. 

A curved line is now drawn through points 1 to 9. Straight 
lines are drawn connecting points F, B, and A. If both sides of the 
pattern have been worked simultaneously, the whole pattern has 
been developed and locks may now be added as shown. 

The straight lines representing the square base are treated 
according to the type of joint adopted for the system of piping, of 
which this "fitting" is a part. 



200 SHEET METAL DRAFTING 

Problem 49 

TRANSITION BETWEEN AN OVAL PIPE AND THE SECOND 
PIECE OF AN ELBOW 

88. Transition between an Oval Pipe and the Second Piece 

of an Elbow. — Figure 284 is constructed by first drawing an eleva- 
tion of the required elbow according to the directions given in 
Chapter III. The first piece of the elbow is then erased and the 
elevation of the transition added. A half-profile is then drawn 
adjacent to the small end of the elbow and divided into eight equal 
parts. These divisions are numbered as shown, and extension 
lines are carried through the elevation until they meet the first 
miter fine, where corresponding numbers are placed at each inter- 
section. 

From the intersections of the miter fine vertical extension lines 
are dropped. These extension lines are crossed by the horizontal 
center line AK as shown in Fig. 286. About this center line the 
plan, Fig. 286, is now drawn. The vertical extension hnes divide 
the circumference of the circle into equal parts. One-half of the 
oval profile may now be equally divided, although the straight fine 
EF may be considered as one space. 

A true section on the miter line should now be developed as 
shown by Fig. 285, in the following manner. Extension lines are 
carried horizontally from points 1 and 9 of Fig. 284, and a new line 
1-9 is drawn parallel to fine 1-9 of Fig. 284. The exact spacings 
of the miter line are transferred to this line, and perpendiculars 
are erected at each point with numbers to correspond. Upon each 
of these perpendiculars and on each side of the line 1-9, a distance 
is laid off equal to the distance from a correspondingly numbered 
point in the half-profile to the center line 1-9 of the half-profile. 
A curved line traced through the intersections thus obtained is a 
true section on miter line 1-9 of Fig. 284. 

Before the base lines for the triangles can be drawn in Fig. 286, 
the order of triangulation must be determined. In a transition of 
this Ivind the elements of the surface must alternate between the 
upper and lower bases in order to have sufl&cient data with which 
to develop the pattern. A standard order of triangulation is 
given below, together with the altitude for each triangle. 

The base lines may now be drawn in Fig. 287 according to this 
order. 



TRIANGULATION OF TRANSITION PIECES 201 




•y 


, i 


"^" 


i\ 


I't'S 


\» 


fi I 


I''*! 


1 lift 


1 1 
1 1 
1 1 


1 1 
1 1 




[r^ 


•\ ^ 


\ 




H 






> 

O 



f- 



Pm 



202 SHEET METAL DRAFTING 

Order of Triangolation for Fig. 286 



Triangles. 


Altitudes. 


Triangles. 


Altitudes. 


Triangles. 


Altitudes. 


Atol 


1 


Dto4 


4 


(? to7 


7 


1 to 5 


1 


4: toE 


4 


1 ioH 


7 


Bio 2 


2 


E to5 


5 


Hto8 


8 


2 toC 


2 


5 toF 


5 


8 to J 


8 


(7 to 3 


3 


F to 6 


6 


J to 9 


9 


3 toD 


3 


6 to G 


6 


9 to A' 


9 



The diagrams of triangles, Figs. 287 and 288, are now con- 
structed by drawing short horizontal lines equal to the base lines 
of Fig. 286 with letters and numbers to correspond. Upon per- 
pendiculars erected at one end of each of these lines the proper 
altitude is placed. The true lengths of the altitudes are plainly 
marked- in Fig. 284. 

The pattern development is started by drawing a straight hne 
and setting off upon it a distance equal to the hypotenuse of triangle 
A to 1. Next in order comes triangle 1-B, so with point 1, Fig. 
289, as a center and with a radius equal to the hypotenuse of 
triangle 1--B, an arc is drawn bearing away from point A. This is 
intersected at B by an arc drawn from point A, with a radius equal 
to line AB of Fig. 286. 

With B as a center and a radius equal to the hypotenuse of 
triangle B-2, an arc is drawn bearing away from point 1. This is 
intersected by an arc drawn from point 1, with a radius equal to 
the distance 1-2 of Fig. 285, thereby establishing point 2 of the 
pattern. In this manner the entire half pattern is developed by 
following the order of triangulation, taking the spaces between the 
figures from Fig. 285, and the spaces between the letters from Fig. 
286. Should the draftsman require a whole pattern, he would 
work both ways from the center line A-1, as the development 
progressed. 



CHAPTER XV 
DEVELOPMENTS BY SECTIONS 



Prob. 
No. 



Job 



Drawing 
Objective. 



Mathematical 
Objective. 



Frustum 

C OME 



OF A 



50 




pr//?c/p/e5 u/fcfer- 
/y/'/?^ ifte study 
of efeve/o/)me/if 
J?y 5ect/o/73. 



51 




Pesigning ifje 
ff'fting. App/y/ng 
ff?e pr//?c/'p/es op 
dpi/e/op/»ent ^ 
sect/ons. 



Oval to 

ROUMO 

"Opfset Boot 



Square, to Round 
Split Header 



sa 




sect/on o/f Hf9 
^erfica/ miter 
^ine. 



Proporf/on/rtg tfre 

p/pes. 
Per c en tag e of 
a/r <^tf>//> «/■«*</. 



Objectives of Problems on Developments by Sections. 
203 



204 SHEET METAL DRAFTING 

Problem 50 
THE FRUSTUM OF A SCALENE CONE 

89. The Frustum of a Scalene Cone. — Triangulation is the 
universal tool of the Sheet Metal Draftsman. Any surface cap- 
able of being developed can be developed by this method. How- 
ever, in the case of the cone and the cyhnder less laborious methods 
are available which are just as accurate. In many problems that 
cannot be classed as parallel line or tapering form developments, 
there is a shorter method known as Development by Sections. 

This method is generally employed in problems where solids 
can be cut into two equal parts by the cutting planes. Figure 290 
shows a frustum of a scalene cone cut by a vertical plane in such a 
manner as to divide it into two equal parts. 

Figure 291 shows one of these halves placed so that the cutting 
plane assumes a horizontal position. If the semicircular ends 
(profiles) were divided into four equal parts, and perpendiculars 
dropped from each of these points to the base lines, a model cut 
along these hues would show the sections as pictured in Fig. ,291. 
It is evident that hues l-A and 5-i? are true length hues, while 
C-4, 4-D, etc., are upper bases of trapezoids and must be developed. 
After the pattern has been developed, a cardboard model of Fig. 
291 made from Fig. 292 and the diagram of sections, Fig. 293, 
should be constructed to aid in the visuahzation of future problems. 

Figure 292 is a plan or top view of the object with half-profiles 
attached to each end. These half-profiles are divided into equal 
parts and extension lines carried to Unes 1-5 and AE as shown. 
The divisions are then numbered and lettered. Before the base 
lines of the sections can be drawn in, the order of sections must be 
determined. The standard adopted for triangulation can still be 
adhered to and the order would read A-2, 2-5, J5-3, 3-C, C-4, 
4-D, and i)-5. Since hues l-A and b-E are true lengths they 
need not be mentioned in the order. Having determined the 
proper order, the base lines on the plan are now drawn in as shown 
in Fig. 292. 

The diagram of sections. Fig. 293, is now constructed by draw- 
ing short horizontal lines equal in length to the several base hues 
of Fig. 292 and with numbers and letters that correspond to the 
order adopted. Perpendicular lines are erected at each end of 



DEVELOPMENTS BY SECTIONS 



205 



these lines. Upon these perpendiculars are set off the lengths of 
the correspondingly numbered extension lines in the half-profiles. 
Attention is called to the fact that points 1 and 5 have no altitudes, 
since they fall on the horizontal plane upon which the entire figure 




o 






& 



rests. Straight lines connecting the points estabhshed upon these 
perpendiculars are the true lengths from which the pattern is 
developed. 

The pattern is started by drawing a Horizontal hne such as line 
1-A of Fig. 294. With A as a center and a radius equal to the 



206 SHEET METAL DRAFTING 

hypotenuse of section A to 2, an arc is drawn bearing away from 
point 1. This is intersected by an arc drawn from point 1, with a 
radius equal to the distance 1-2 of the profile. This establishes 
point 2. With point 2 as a center and a radius equal to the upper 
base of section 2-B, an arc is drawn bearing away from point A. 
This is intersected at B by an arc drawn from point A, with a radius 
equal to distance AB oi the profile. In this manner all the 
points of the pattern are located in the order previously adopted. 
Curved lines passing through these points give the half pattern 
for the entire frustum. 



DEVELOPMENTS BY SECTIONS 207 

Problem 51 
CENTER OFFSET BOOT 

90. Center Offset Boot. — The pictorial drawing shows the 
conditions under which a center offset boot is used. The oval wall 
stack or riser descending from the upper stories generally comes in 
the center of the stringer. Enough of the stringer is cut away at 
an angle of 45° to permit the boot to be connected. 

As in the case of the scalene cone, this fitting can be cut by a 
vertical plane so as to form two equal parts. Figure 295 is a plan 
view of one of these parts. In drawing this view, care must be 
taken to get the miter lines at the correct angle for a two-piece 45° 
elbow. 

The half-profiles are then drawn in their relative positions and 
divided into equal parts. It has been observed that in treating 
oval profiles the straight sides of the oval are never divided. The 
divisions of the curved portions are numbered and lettered and 
extension fines carried from each division to the miter lines. 

From this view the pattern of the round end and also the oval 
end of the fitting can be developed according to the rules given in 
Chapter III. 

The center or transition piece of the fitting is developed by 
means of sections. Figure 296 shows this transition moved to one 
side in order to avoid a confusion of lines. The intersections of 
both miter lines are also transferred. Perpencficulars are erected 
at each intersection and the distances from points B, C, and D, 
to the center fine of the half-profile, Fig. 295, set off on corre- 
sponding fines at one end of Fig. 296, and distances from points 
2, 3, 4, 5, and 6 to the center line of the half-profile set off on 
corresponding perpendiculars at the other end of Fig. 296. Curved 
fines traced through these points give the true sections on the 
miter fines. 

The diagram of sections, Fig. 299, is now constructed by draw- 
ing horizontal lines equal in length to the base fines in Fig. 296. 
Perpendiculars are erected at each end of these fines and lettered 
and numbered to correspond to the base lines. Upon these per- 
pendiculars are set off distances equal to the length of correspond- 
ingly numbered and lettered lines in the true sections of Fig. 296. 
The lengths of straight lines connecting these points are the true 



208 



8HEET METAL DRAFTING 



lengths of the lines needed to develop the pattern. The section 
A to 2 has for its base line the true length line A to 1 of Fig. 296; 
therefore, both base and hypotenuse of this section are used in the 
development of the pattern. This is also true of section E to 7. 
The pattern, Fig. 300, is started by drawing a vertical Hne 




pq 



U 



1^ 



equal in length to Hne A-1 of Fig. 296. With point A as a center 
and a radius equal to the hypotenuse of section A to 2, an arc is 
drawn bearing away from point 1. This is intersected by an arc 
drawn from point 1 with a radius equal to Hne 1-2 of the true 
section. Fig. 296. This estabHshes point 2. 



DEVELOPMENTS BY SECTIONS 209 

With point 2 as a center and a radius equal to the upper base of 
section 2 to B, an arc is drawn bearing away from point A. This is 
intersected by an arc drawn from point A with a radius equal to 
distance AB oi the true section, Fig. 296, thereby establishing 
point B. In this manner all of the points of the pattern are fixed 
and the curved and straight lines of the pattern drawn in. 

Care should be observed with regard to these items: 

(a) The spacing between the points of the pattern must be 
taken from corresponding spaces in the true sections. 

(6) Distances 1-2 and 7-6 are greater than any of the other 
spaces and must be connected by straight lines . 

(c) A chisel point must be used in the compass to assure fine 
lines without which the necessary accuracy cannot be attained. 
The whole pattern may be produced by copying on the other side 
of fine A-1 the half which has already been drawn. 

The pattern may be checked for accuracy by ascertaining 
whether or not the angle E-7-6 is a right angle. If this angle is 
of more or less than 90°, the pattern is incorrect. In this connec- 
tion it should be observed that a slight error in any measurement 
will throw the whole pattern "out of true." 



210 SHEET METAL DRAFTING 

Problem 52 
SQUARE TO ROUND SPLIT HEADER 

91. Square to Round Split Header.— This type of fitting is 
often used where a fan with a rectangular outlet must supply two 
round pipes running in different directions. 

This problem presents a case wherein development both by 
triangulation and by sections may be employed in order to obtain 
the pattern. 

The pattern is in reality two square to round transitions miter- 
ing upon each other. Where the two transitions come together a 
miter line is produced. A true section on this miter line must be 
developed. The practice of assuming a section, common to some 
drafting rooms, results in a more or less distorted fitting, accord- 
ing to the experience the draftsman has had in designing such fit- 
tings. The workman in forming and assembling the fitting has 
difficulty in that he must compel the assembly to take an unnatural 
shape. 

The plan, Fig. 301, is first drawn according to the dimensions 
taken at the job. The profiles are then divided into the same 
number of equal spaces. After the order of triangulation has been 
determined, base fines are carried to the corners M and N of the 
rectangle. 

The elevation is drawn in its proper location by means of exten- 
sion fines carried from the plan. The miter fine is drawn in both 
plan and elevation and should pass through the intersections of 
the base lines, Fig. 301, at points A, B,C, D, and E; it should also 
pass through the elevation of the elements, Fig. 302, at points F, 
G, and H. 

A true section on this miter line should now be developed by 
drawing a horizontal line equal in length to twice the distance AE 
of Fig. 301. The center point of this line. Fig. 303, should be 
lettered A, and a perpendicular center line erected. Upon each 
side of this center, points B, C, D, and E should be located exactly 
as they appear on the miter fine of Fig. 301. Perpendiculars are 
now erected at each of these points. Point B in Fig. 301 falls on 
fine A^'-l, and point H falls on line A^-1 of Fig. 302; therefore, the 
perpendicular height of point H from the base line of Fig. 302 
should be set off on perpendicular B of Fig. 303. For the same 



DEVELOPMENTS BY SECTIONS 



211 



reasons point G should be located on perpendicular C and point F 
on perpendicular D. A curved line traced through these points 
gives a true section on the miter Kne. 




//5 


Xw ^ 


/ / o 
/ ° 




/ o 


/ ^s ^a 












/ ^ 


'--. 1 2I-0 




, ""'-- u. 5 


mZ ii. 


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\ / e° 


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\ '" ^ij 


\ \ ^ 


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ax""' 


K r 
HO 


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o 



3 

J/2 





h~i|o >o apn4i4{o«nj^« 



Because of the intersections of the miter hne it will now be 
necessary to revise the order of triangulation originally adopted 
when the fitting was considered as two separate square to rounds. 
The original and revised orders are as follows: 



212 



SHEET METAL DRAFTING 



Original Order. 


Revised Order. 


A^ to 1 
N to 2 
iV to3 
3 toM 
M to4 
M to 5 
5 to A' 


N to B and 1 
A^ to C and 2 
A^ to D and 3 
3 to iW 
M to 4 
M to 5 
5 to K 



As will readily be seen the revision is based on the fact that 
lines N-l, N-2, and A^-3 of Fig. 301 cross the miter line at 
points B, C, and D. 

The diagram of triangles, Fig. 304, is now constructed by draw- 
ing horizontal Unes equal in length to the base Hnes in Fig. 301. 
Perpendicular lines are erected at each end of these Unes and also 
at points B, C, and D. Since both the upper and the lower planes 
of the fitting are parallel there will be but one altitude to the tri- 
angles. This altitude is shown in Fig. 302 and should be placed 
on perpendiculars 1, 2, 3, M, 4, 5, and K of Fig. 304. The hypot- 
enuses of these triangles may now be drawn and points H, G, and 
F located by the intersection of perpendiculars B, C, and D with 
the respective hypotenuses. 

The pattern, Fig. 305, is started with a horizontal line equal to 
line N-O of Fig. 301. With N and as centers and a radius equal 
to the hypotenuse of triangle iV to 1, intersecting arcs are drawn 
above the line, thereby locating point 1. Since both sides of the 
fitting are equal the pattern may be developed from points N and 
simultaneously. The remainder of the pattern is developed 
exactly as was the square to round transition of Chapter XIII. 

After the entire pattern has been developed the miter cut is 
developed as follows: Point E is located on fine MN of Fig. 305 
exactly as it appears on line MN of Fig. 301. The hypotenuse of 
triangle A^ to D is placed on line A^-3 of Fig. 305, thereby locating 
point F. Placing the hypotenuse of triangle A to C on line A^-2 
of Fig. 305 locates point G, and the hypotenuse of triangle A" to 
B placed on line A'-l locates point H. 

The miter cut is drawn with straight Hnes between points E 
and F and curved Unes connecting points F, G, and H of Fig. 305. 
This completes the pattern except for locks and riveting laps. 



DEVELOPMENTS BY SECTIONS 



213 



In case the round pipes are of unequal diameter the order of 
triangulation is altered somewhat, but the general method of pro- 
cedure remains the same. The true section on the miter line is 
developed by considering the fitting as a transition between the 
round pipe of larger diameter and the whole of the rectangular 
base. This section is used for the development of both pieces 
and the order would read: 



Order. 


Proper Altitudes. 


Spaces obtained from. 




• K to 5 


True altitude of all triangles 


Circular profile in 






as shown in Fig. 301 


Fig. 301 




5 toM 


True altitude of all triangles 


Base of rectangle in 






as shown in Fig. 301 


Fig. 301 


Triangles • 


M to4 


True altitude of all triangles 


Circular profile in 




as shown in Fig. 302 


Fig. 301 




Mto3 


True altitude of all triangles 


Circular profile in 






as shown in Fig. 302 


Fig. ,301 




d toE 


True altitude of all triangles 


Base of rectangle in 






as shown in Fig. 302 


Fig. 301 




3 toD 


True altitude and alt. of F, 


E to F in true section, 






Fig. 303 


Fig. 303 




D to2 


True altitude and alt. of F, 


3 to 2 of circular pro- 






Fig. 303 


file, Fig. 301 




2 toG 


True altitude and alt. of G, 


F to G in true sec- 


Sections 




Fig. 303 


tion, Fig. 303 


G tol 


True altitude and alt. of G, 


2 to 1 in circular pro- 






Fig. 303 


file. Fig. 301 




1 to 5 


True altitude and alt. of H, 


G to 27 in true sec- 






Fig. 303 


tion, Fig. 303 




1 to^ 


True altitude and alt. of H, 


fl^ to A in true sec- 






Fig. 303 


tion. Fig. 303 



It is required that the pattern be redeveloped according to the 
above order and the results compared with the first development. 

92. Related Mathematics on Split Headers. — The ques- 
tion as to how large the branch pipes can or should be made 
often arises in fan or blower design. 

The following factors enter into the consideration of such 
questions : 

(a) What percentage of the total volume available must be 
delivered in a given direction? 



214 SHEET METAL DRAFTING 

(6) How many and what kinds of machines are to be served by 
the branch pipes? 

(c) Losses caused by friction, and the effects upon static and 
velocity pressures caused by changes in cross-sectional area of the 
duct. 

Item (c) is a matter that largely concerns the engineer although 
the sheet metal worker would do well to have some understanding 
of these things. 

Items (a) and (6) , however, are matters of common arithmetic 
and the following problems are based on them. 

Problem 5M.— The fan outlet measures 19" X 35|". One 
branch of a split header is equal in area to 69 per cent of the area 
of the outlet. If both branch pipes are round, what are their 
diameters? 

Problem 52B. — A 20" pipe carries 75 per cent of the air from a 
spHt header. What is the diameter of the other round pipe? 

Problem 52C. — A split header has one 15" and one 18" branch. 
What will be the dimensions of the rectangular opening 19" wide 
that will accommodate these two branches? 

Problem 52D. — On one side of a fan the machines to be served 
require four 6", three 4", and two 3" pipes, while those on the 
other side require six 8", three 6", and two 4" pipes. What will 
be the area of the branch pipes that are needed to serve these 
machines? (Hint; Loss of head not considered.) 



CHAPTER XVI 
DEVELOPED AND EXTENDED SECTIONS 



Proh 
No. 


Job 


Oravs/ing 
Objective 


Mathematical 
OBJECTWE 


53 




^ 


Design rr?^ Mmg. 
Deve/op/n^sec - 
f/o/7s 0/7 ^ar/oc/s 
m/^er //hes. 




Oval TO Rou 
Elbow 


NO 


54 


lii 


1 


Design of 
f/ff/ngs 




Breeches 


55 


fr 


sect/'o/7s. 





Objectives of Problems on Developed and Extended Sections. 

215 



216 SHEET METAL DRAFTING 

Problem 53 
OVAL TO ROUND ELBOW 

93. Oval to Round Elbow. — Figure 306 is an elevation of an 
elbow whose first piece is that of a five-piece round pipe elbow, 
whose last piece is that of a five-piece oval elbow, and whose inter- 
mediate pieces are transitions which gradually convert the round 
pipe to the oval pipe. 

This elevation is constructed by first drawing an angle (angle 
AJF) equal to the required angle of the elbow. The throat radius 
is then laid off on the base fine and the arc of the throat drawn 
in. This arc is then divided and the miter fines of the elbow are 
drawn as though the elbow were a regular round pipe fitting such 
as described in Chapter III. Lines AB, GH, ON, and FE are 
drawn at right angles to the fines AG and OF, thereby complet- 
ing the elevation of the first and last pieces. The throats of the 
second, third, and fourth pieces are drawn tangent to the arc as 
described in Chapter III. 

The length of miter fines CK and DAI are now determined in 
order to complete the outfine of the back. Since the reduction 
must be gradual, an equal amount must be subtracted from each 
succeeding miter line. A short and convenient method of finding 
these lengths is shown in Fig. 313. Four parafiel lines are drawn 
and the lengths of lines BH and EN placed upon them so that 
point E is directly under point B. The difference in length be- 
tween these two fines is divided into three equal parts and perpen- 
diculars dropped from each of these points to the other parallel 
lines, thereby estabhshing the lengths of miter fines CK and DM. 

These lengths are now placed on the proper miter fines in Fig. 
306, and the outline of the back of the elbow is completed, by con- 
necting points B, C, D, and E with straight lines. 

Each miter line as it passes through the elevation is divided 
into two equal parts, thus locating center points P, R, S, and T. 
The center fine of the elevation may then be drawn by connecting 
these points. 

The oval end of the elbow has two flat surfaces while the round 
end has none. Consequently, these flat surfaces must gradually 
diminish in width untfl they disappear upon reaching the round 
end of the elbow. 



DEVELOPED AND EXTENDED SECTIONS 217 

In order to develop the patterns it is necessary to know the 
exact widths of these flat surfaces at each cutting plane, and in 
order to gain this information stretchouts must be made as shown 
by Figs. 307, 308, and 309. 




I o 





z 




J 






^ 


g 


1 


6 


I 


7 


s 


J 


t 




i 


s 




a 


x 


Ji 


► 


10 i 


1 

s 


1 




i 




ti. ^ 






































z 


X 


£ 


I 


















■0* 


1 
o 


1 
u 


1 


1 

3 


> 


1 


i 


> 


f 


1 


X 


« 


> 


y 


^ 


I 


t 




fe f:^ 




03 

o 



Upon any vertical line a stretchout of the throat of Fig. 306 is 
made as shown by points 0, N, M, K, H, and G of Fig. 307. 
Through each of these points perpendicular Hnes are drawn. 
Upon Unes and N one-half of the straight line in the oval profile 
must be laid down on each side of points and N. Straight lines 



218 SHEET METAL DRAFTING 

connecting these points produce a rectangle, showing the true 
shape of the flat surface in the throat of the last piece of the elbow. 
If this surface were allowed to taper, the oval pipe to which the 
elbow is joined would not fit properly. Since the flat surface is to 
disappear at the throat of the first piece, straight lines may be 
drawn from the extremities of line N to the point H, thus establish- 
ing widths at M and K. 

The widths of the flat surfaces on the back of the elbow, as 
shown by Fig. 308, are developed in exactly the same manner, 
and the description given above may be used again by substituting 
the word back and the letters that correspond. 

Since the major axis (long diameter) of the oval is greater 
than the diameter of the circular profile a gradual increase must 
also be made in the major axes of the sections formed by the sev- 
eral miter lines. A stretchout of the center line of Fig. 306 is made 
as shown by Fig. 309. On the horizontal line T the long diam- 
eter of the oval profile is placed (one-half on each side of point T"), 
while on line P one-half of the diameter of the round profile is 
placed. Straight lines are then drawn connecting these points, 
thereby establishing the major axes of the sections formed by 
miter lines DM and CK. 

The patterns for the first and last pieces can be drawn by the 
method described in Chapter III since they are pieces of regular 
five-piece elbows. 

The intermediate pieces of the elbow must be developed sepa- 
rately, either by triangulation or sectional development. The third 
piece of the elbow has been selected for treatment in this descrip- 
tion, but the second and fourth pieces are developed in exactly 
the same manner. 

Figure 310 shows the third piece removed from Fig, 306, in 
order to avoid confusion of lines. At the center points R and S per- 
pendicular center lines are erected. Upon the center line at S one- 
half of the major axis of section DM, Fig. 309, is set off; and 
upon the center line at R, one-half of the major axis of section CK, 
Fig. 309, is set off. These points are lettered 4 and R as shown. 
Perpendiculars are now erected at points D, M, K, and C. Upon 
these lines the following lengths are placed: line D-1 is made 
equal to one-half of line D in Fig. 308; line C-U one-half of line C 
in Fig. 308; line M-7 one-half of line M in Fig. 307; and fine K-Z 
one-half of line K in Fig. 307. Arcs are now drawn connecting 



DEVELOPED AND EXTENDED SECTIONS 219 

points 1, 4, and 7, and also points U, R, and Z. Sections are thus 
formed on cutting planes DM and CK, and while these are not 
absolutely true sections, as defined by the laws of projection, they 
are near enough for all practical purposes and can be constructed 
in much less time. 

The arcs of these sections are divided into equal spaces and the 
divisions given numbers or letters as shown in Fig. 310. Perpen- 
dicular lines are dropped from each division until they intersect 
Knes DM and CK. 

The order of sections is now decided upon as shown by the 
order given upon the drawing. The base lines for the sections 
may or may not be drawn in Fig. 310, according to the amount of 
dependence to be placed on their guidance. 

The diagram of sections, Fig. 311, shows the condensed form 
in which the experienced draftsman usually develops this feature 
of the problem. All of the altitudes of points in the profile of one 
end of the transition are placed upon a vertical line. From the 
intersection of this vertical line with a horizontal base line all of 
the base hues from Fig. 310 are measured. Above this horizontal 
line, other horizontal hues are drawn at distances representing the 
altitudes of the points in profile of the other end of the transition. 

From Fig. 310 base hnes are laid down upon the horizontal line 
of Fig. 311, and perpendicular lines are erected to the proper alti- 
tude. The upper base of the section can be measured with the 
dividers and used in developing the pattern. 

Supposing the true length of the upper base of Section i? to 5 is 
desired. Starting at the short hne labeled R-5, the vertical line 
is followed upwards until it meets the horizontal line labeled Alt. 
of R. The distance between this point and point 5 on the vertical 
Une at the left of the diagram is the required distance. 

It sometimes happens that two base lines have the same 
lengths as is the case with C-l and 1-U. To find the true lengths, 
starting at C-l, the vertical line is followed upwards until it meets 
the base hne of the diagram (points C and K having no altitude) 
and from this point the distance to point 1 on the vertical line at 
the left of the diagram is measured. Again starting at 1-t/ the 
same vertical line is followed upwards until it meets the horizontal 
line representing the altitude of U. The distance from this point 
to point 1 at the left of the diagram is the true length sought. 

Starting with a vertical hne. Fig, 312, upon which the length 



220 SHEET METAL DRAFTING 

of line DC of Fig. 310 has been placed, the pattern is developed in 
the usual manner by following the order of development, by sec- 
tions, which has already been determined. The flat surfaces 
should be marked to aid the workman in forming the metal. The 
whole pattern may be produced by copying the pattern which has 
been developed on the other side of Une DC. 



DEVELOPED AND EXTENDED SECTIONS 221 



Problem 54 



BREECHES 



94. Breeches. — Breeches is the trade name given to a transi- 
tion between two round pipes and an oval or round pipe of larger 
diameter. The plan, Fig. 314, is that of a transition between two 
round pipes of unequal diameters, and an oval pipe. The plan is 
first drawn showing the branch pipes in their proper location. A 
horizontal and a vertical center line are then drawn in the oval 
profile. As will be seen upon examination the horizontal center 
line divides the figure into two equal parts; therefore, it is capable 
of being developed by sections. The vertical center line of the 
oval will be used as the miter line between the two branches. 

The profiles of Fig. 314 are divided, as shown, after which 
extension lines are dropped and an elevation, Fig. 315, constructed. 
Each point in the profiles should be properly located by extension 
lines in the elevation. 

The order of development should now be decided upon. As 
in the case of the split header. Chapter XV, the fitting should be 
considered from the standpoint of two separate transitions between 
oval and round pipes. In this problem the large branch is treated 
first and the order of development determined as follows : 

Order of Development for Large Branch 









Intersection with Miter Line: 


Triangles 


Triangles 


Triangles 


Base line 3 to 


5 toG 


F to3 


2 tofi 


Base line C to N 


G*to4 


^ toE 


B tol 


Base line 2 to if 


4 toF 


3 toC 


1 to A 


Base line B to K 




Cto2 




Base line 1 to H 



The base Unes corresponding to this order are now drawn in 
Fig. 314 and their points of intersection with the miter Hne indi- 
cated by letters 0, N, M, K, and H as shown. These are given 
in the fourth column of the above table, and should be placed in 
the diagram of triangles, Fig. 318, exactly as indicated; that is, 
base line 3 to should be measured from point 3 in the diagram 
of triangles and not from point C. 

The diagram of triangles is now drawn by taking the base Hnes 



222 



SHEET METAL DRAFTING 



from Fig. 314 in the order given above. Perpendiculars are 
erected as shown and since the planes of the transition are parallel 
all triangles will have the same altitude. This altitude is shown 




cq 



^ 



by line 6-^ of Fig. 315. Perpendiculars erected at points 0, N, 
M, K, and H estabhsh the position of these points on the hypote- 



DEVELOPED AND EXTENDED SECTIONS 223 

nuses of their respective triangles and enable the true lengths of 
these lines to be measured. 

An elevation of these elements is now drawn in Fig. 315. 
Where these elements intersect the miter line the elevation of 
points 0, N, M, K, and H will be established. 

A true section on the miter line, Fig. 316, is now constructed by 
transferring the spacing of miter Kne HD in Fig. 314 to any hori- 
zontal line. Perpendiculars are erected at each of these points and 
corresponding altitudes, taken from Fig. 315, placed on them. 
Lines connecting the points thus located constitute a true half 
section on the miter line. The whole section may be produced, if 
desired, by copying on the other side of center line H, the half 
already developed. 

The pattern for the large branch is started by placing on any 
vertical line a distance equal to the hypotenuse of triangle 5 to G. 
The pattern is developed in the usual manner by following the 
order of development previously determined upon. 

After the pattern for a complete transition has been developed 
the miter cut is drawn in as follows : The distance from point C 
to point on the hypotenuse of triangle 3-0-C is laid off from 
point C on line C-3 of Fig. 320. Similarly, the distances C to N, 
B to M, B to K, and Aio H are laid off from points C, B, and A on 
their corresponding Hnes in Fig. 320. A curved line passing 
through these points gives the miter cut of the pattern. Partic- 
ular attention must be given to keeping the direction in which 
these measurements are taken, the same in the plan, in the dia- 
gram of triangles, and in the pattern. 

In order to miter the small branch with the large branch, the 
same section must be used on the miter line. Figure 317 shows a 
half plan of the small branch removed from Fig. 314 in order to 
avoid confusion of lines. All of the intersections have been trans- 
ferred and it is necessary to consider a new order of development 
which is given below. 

Order of Development for Small Branch 



True Len 


gth 6 to A 


Triangle 8 to C 


Section 9 to M 


Triangle 


A to 7 


Triangle 8 to D 


Section 9 to K 


Triangle 


1 ioB 


Triangle D to 9 


Section K to 10 


Triangle 


£to8 


Section 9 to 
Section 9 to A^" 


Section 10 to E 



224 SHEET METAL DRAFTING 

The diagram of triangles and sections, Fig. 319, is now con- 
structed by taking the base hnes from Fig. 317. The triangles 
have a common altitude equal to hnes Q-A of Fig. 315. One of 
the perpendiculars in each section also has this altitude, but the 
altitudes of points 0, N, M, K, and H are made to correspond to 
the altitudes of similar points in the true section. Fig. 316. 

The pattern. Fig. 321, is started by placing upon any vertical 
line a distance equal to the true length hne Q-A. The develop- 
ment proceeds in the usual manner until completed. Spaces A to 
D are taken from corresponding spaces in Fig. 317, as are also 
spaces 6 to 10, Spaces D to H, however, are taken from corre- 
sponding spaces in the true section. Fig. 316. The development 
may be copied on the other side of line 6-^ in order to obtain the 
full pattern. 

Necessary laps for riveting the pieces together must be added 
to these patterns. 



DEVELOPED AND EXTENDED SECTIONS 225 

Problem 55 
COAL HOD 

95. The Coal Hod. — The coal hod is made in many different 
designs, and since there is no standard the manufacturer draws the 
design to suit his own ideas. 

Figure 324 shows a half -profile of a coal hod, and since the 
center line divides the object into two equal parts the pattern 
can be developed by sections. 

Figure 322 shows the plan of the coal hod after it has been cut 
by an imaginary plane and so placed that the cutting plane assumes 
a horizontal position. 

Figure 323 is a half -profile of the bottom located in its relative 
position by means of extension lines. 

Both profiles should be divided into equal spaces and each divi- 
sion numbered or lettered as shown. It is evident that line 1-10 
of Fig. 324 is longer than line 1-10 of Fig. 322 (since a straight line 
is the shortest distance between two points). Because of this fact 
the distances between points in the profile, Fig. 324, are not true 
lengths. 

Figure 325, which is sometimes called an extended section, must 
be developed in order to ascertain the true distances between these 
points. A tangent parallel to the bottom line of the coal hod is 
drawn through the plan as shown in Fig. 322. Since point 4 is 
the lowest part of the curve the tangent must pass through this 
point. Extension lines are then carried from each point in the 
curve until they meet the tangent as shown in Fig. 322. 

Upon any straight line, Fig. 325, the exact spacing between 
points 1 to 9 of Fig. 324 are laid off. Perpendiculars are erected 
at each point, and upon each perpendicular a distance equal to 
that from a correspondingly numbered point in the curve to the 
tangent of Fig. 322 is set off. 

A curve traced through the points thus obtained will give the 
exact distance between these points, and these must be used in 
developing the pattern. 

The order of sections must now be determined, such as will be 
found on the drawing, and a diagram of sections, Fig. 326, con- 
structed. The base lines for this diagram are taken from the plan. 
Fig. 322. The altitudes 1 to 10 are taken from Fig. 324, and the 
altitudes AtoK from Fig. 323. 



226 



SHEET METAL DRAFTING 



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DEVELOPED AND EXTENDED SECTIONS 227 

The pattern is started by placing upon any straight Hne a dis- 
tance equal to line 1-A of Fig. 322, this being a true length Hne 
since it rests on the horizontal plane. From point A point 2 is 
estabHshed and the pattern proceeds in the usual manner, using 
Fig. 325 for the spacing between the numbered points, and Fig. 
323 for the spacing between the lettered points. The space be- 
tween points 9 and 10 is taken from Fig. 324, and the line K-10 
of the pattern is taken from line iv-10 of Fig. 322 since it rests on 
the horizontal cutting plane and is, therefore, a true length. 

The coal hod is generally made from two pieces of metal, locks 
being added parallel to lines A-1 and K-10 of pattern. However, 
if it is desired that the object be made from one piece, the develop- 
ment may be copied on the other side of Une K-10 in Fig. 327. 



228 



SHEET METAL DRAFTING 



TABLE A 

U. S. Standard Gage for Sheet and Plate Iron and Steel, 1893 



Number 


Approximate 
Thickness in 


Approximate 
Thickness in 


Weight per 


Weight per 


of 
Gage. 


Fractions of 
an Inch. 


Decimal Parts 
of an Inch. 


Square Foot 
in Ounces. 


Square Foot 
in Pounds. 


0000000 


1/2 


0.5 


320 


20 


000000 


15/32 


0,4688 


300 


18.75 


00000 


7/16 


0.4375 


280 


17.50 


0000 


13/32 


0.4063 


260 


16.25 


000 


3/8 


0.375 


240 


15 


00 


11/32 


0.3438 


220 


13.75 





5/16 


0.3125 


200 


12 . 50 


1 


9/32 


0.2813 


180 


11.25 


2 


17/64 


0.2656 


170 


10.625 


3 


1/4 


0.25 


160 


10 


4 


15/64 


0.2344 


150 


9.375 


5 


7/32 


0.2188 


140 


8.75 


6 


13/64 


0.2031 


130 


8.125 


7 


3/16 


0.1875 


120 


7.5 


8 


11/64 


0.1719 


110 


6.875 


9 


5/32 


0.1563 


100 


6.25 


10 


9/64 


0.1406 


90 


5.625 


11 


1/8 


0.125 


80 


5 


12 


7/64 


0.1094 


70 


4.375 


13 


3/32 


0.0938 


60 


3.75 


14 


5/64 


0.0781 


50 


3.125 


15 


9/128 


0.0703 


45 


2.813 


■16 


1/16 


0.0625 


40 


2.5 


17 


9/160 


0.0563 


36 


2.25 


18 


1/20 


0.05 


32 


2 


19 


7/160 


0.0438 


28 


1.75 


20 


3/80 


0.0375 


24 


1.50 


21 


11/320 


0.0344 


22 


1.375 


22 


1/32 


0.0313 


20 


1.25 


23 


9/320 


0.0281 


18 


1 . 125 


24 


1/40 


0.025 


16 


1 


25 


7/320 


0.0219 


14 


0.875 


26 


3/160 


0.0188 


12 


0.75 


27 


11/640 


0.0172 


11 


0.688 


28 


1/64 


0.0156 


10 


0.625 


29 


9/640 


0.0141 


9 


0.563 


30 


1/80 


0.0125 


8 


0.5 


31 


7/640 


0.0109 


7 


0.438 


32 


13/1280 


0.0102 


6| 


0.406 


33 


3/320 


0.0094 


6 


0.375 


34 


11/1280 


0.0086 


5J 


0.344 


35 


5/640 


0.0078 


5 


0.313 


36 


9/1280 


0.007 


4i 


0.281 


37 


17/2560 


0.0066 


4| 


0.266 


38 


1/160 


0.0063 


4 


0.25 



INDEX 



Anvil, mechanical drawing of, 2 
Apron for conical roof flange, 93 
Area of circle, 36 

cylinder, lateral, 32 

frustum, 91 

lateral, of cone, 82 

oval, 193 

rectangle, 16 

sector, 82 

trapezoid, 23 
Areas, equivalent, 193 
Ash barrel, 110 

quantity production, 110 

related mathematics, 117 
Ash pan with semicircular back, 

related mathematics, 155 
Ash sifter, rotary, 156 

related mathematics, 160 
Atomizing sprayer, 148 

related mathematics, 150 
Axis of cone, 80 

Backset method, 63 
Barrel, ash, 110 

related mathematics, 117 
Bending lines, 7 
Board, drawing, 3 
Boat pump, 125 

related mathematics, 127 
Boot, center offset, 207 
Border lines, 7 
Boxes, register, 143 

related mathematics, 145 
Bread pan, iron, 21 

related mathematics, 23 
Breeches, 221 



153 



Candy pan, 17 

related mathematics, 19 
Can, garbage, 41 

related mathematics, 42 
Can, sprinkling, 120 

related mathematics, 123 
Cap, window, 182 
Center line radius, 61 
Center lines, 8 
Center offset boot, 207 
Chimney tube, galvanized, 30 

related mathematics, 32 
Circle, area of, 36 

circumference of, 32 

diameter, from area, 43 

filing, 10 
Circumference of circle, 32 
Cleat, sheet metal, 12 

related mathematics, 13 
Coal hod, 225 

Combination of various solids, 147 
Conductor head, 177 

related mathematics, 178 
Cone, axis of, 80 

frustums of, 161 

profile of, 80 

revolution of, 79 

right, 80- 

scalene, 186 

scalene, frustum of, 204 

triangulation of scalene, 185 
Conical flower holder, 80 

related mathematics, 82 
Conical roof flange, 89 

related mathematics, 91 



229 



230 



INDEX 



Cover, pitch top, 83 

related mathematics, 85 
Cubic inches in gallon, 40 
Cup, half -pint, 34 

related mathematics, 36 
Cup strainer, 162 

related mathematics, 164 
Curved elbow in rectangular pipe, 106 

related mathematics, 106 
Cutting planes, 119 
Cylinder, cut by planes, 45 

intersecting, 65 

lateral area of, 32 

volume of, 40 

wired, 29 

Definition, miter line, 46 
Developed and extended sections, 215 
Developments by sections, 203 
Diagonal offset, 103 

related mathematics, 105 
Diameter of circle, 43 
Diameters, solids of revolution, 98 
Dimension lines, 8 
Dimensions, fuU size, 41 

over-all, 16 

witness, 42 
Dipper, short handled, 165 

related mathematics, 167 
Drafting, sheet metal, 1 
Drawing board, 3 
Drawing instruments, 3 
Drawings, mechanical, 2 

pictorial, 1 
Dripping or roasting pan, 136 

related mathematics, 138 

Elbows, 

curved, in rectangular pipe, 106 

related mathematics, 106 
four-piece, 90°, 55 
long radius, 57 

related mathematics, 58 
oval to round, 216 
three-piece rectangular, 96 

related mathematics, 97 
two-piece, 49 
two-piece, 60°, 52 



Elbows, weight of, 61 
Elements of surface, 80 
Extension lines, 8 

Extended sections, developed and, 
215 

Face miters, 173, 180 
Figures, rectilinear, 1 
Filing circles, 10 
Flange, conical roof, 89 

apron for conical roof, 93 

related mathematics, 91 
Flange, roof, 129 
Flaring pan, rectangular, 140 

related mathematics, 142 
Flower holder, conical, 80 

related mathematics, 82 
Four-piece 90° elbow, 55 
Frustum, area of, 91 
Frustums of cones, 161 

rectangular pyramids, 135 

scalene cone, 204 

volume of, 167 

Gallon, cubic inches in, 40 
Galvanized chimney tube, 30 

related mathematics, 32 
Galvanized match box, 14 

related mathematics, 16 
Garbage can, 41 

related mathematics, 42 

witness dimensions, 42 
Girth, 32 

Half-pint cup, 34 

related mathematics, 36 
Hammer lock, 106 
Head, conductor, 177 

related mathematics, 178 
Header, square to round split, 210 

related mathematics, 213 
Hod, coal, 225 
Horizontal lines, 4 

Index of problems, 235 
Instruments, drawing, 3 
Intersecting cylinders, 65 
rectangular prisms, 95 



INDEX 



231 



Iron bread pan, 21 

related mathematics, 23 

Lateral area, of cylinder, 32 

of cone, 82 
Layout of cleat, 12 
Lettering, 9 
Lines, bending, 7 

border, 7 

center, 8 

dimension, 8 

extension, 8 

horizontal, 4 

object or projection, 7 

vertical, 5 
Liquid measures, 169 

related mathematics, 171 

standard sizes for, 171 
Lock, hammer, 106 
Long radius elbow, 57 

- related mathematics, 58 

Match box, galvanized, 14 
related mathematics, 16 

Materials, schedule of, 150 

Measures, liquid, 169 

related mathematics, 171 

Mechanical drawings, 1 

Method, backset, 63 

Miter line, definition of, 46 
rise of, problem on, 58 

Miters, face, 180 

Miter, square return, 174 

Miters, return and face, 173 

Moulding, rake, 184 

Objectives of problems, 

combinations of various solids, 
147 

cones of revolution, 79 

cylinders cut by planes, 45 

developed and extended sections, 
215 

developments by sections, 203 

frustums of cone, 161 

frustums of rectangular pyra- 
mids, 135 

intersecting cylinders, 65 



Objectives of problems, 

intersecting rectangular prisms, 
95 

quantity production, 109 

rectilinear figures, 11 

return and face miters, 173 

sections formed by cutting 
planes, 119 

triangulation of scalene cone, 185 

triangulation of transition piece, 
195 

wired cylinders, 29 
Object lines, 7 
Offset, center boot, 207 
Offset, diagonal, 103 

related mathematics, 105 
Offset, rectangular pipe, 100 

related mathematics, 102 
Orthographic projection, 2 
Oval, area of, 193 
Oval pipe and second piece of elbow, 

transition between, 200 
Oval to round elbow, 216 
Oval to round transition, 191 

related mathematics, 193 
Over-all dimensions, 16 

Pail, painter's, 38 

related mathematics, 40 
Painter's pail, 38 

related mathematics, 40 
Pan, ash, with semicircular back, 153 

related mathematics, 155 
Pan, bread, iron, 21 

related mathematics, 23 
Pan, candy, 17 

related mathematics, 19 
Pan, roasting or dripping, 136 

related mathematics, 138 
Pans, rectangular flaring, 140 

related mathematics, 142 
Parallel line developments, rules for, 

75 
Parer, vegetable, 86 
Pencil, 6 

Pictorial drawing, 1 _ 
Pipe, curved elbow in rectangular, 106 

related mathematics, 106 



232 



INDEX 



Pipe offset, rectangular, 100 
related mathematics, 102 

Pipe, standard cuts of, 61 

Pitch top cover, 83 

related mathematics, 85 

Planes, cylinders cut by, 45 

Prisms, intersecting rectangular, 95 

Problems, index of, 235 

Projection lines, 7 

Projections, orthographic, 2 

Pump, boat, 125 

related mathematics, 127 

Pyramids, rectangular, frustums of, 
135 

Quantity production, ash barrel, 110 

Rake moulding, 184 
Rectangle, area of, 16 
Rectangular elbow, three-piece, 96 

related mathematics, 97 
Rectangular flaring pans, 140 

related mathematics, 142 
Rectangular pipe, curved elbow in, 

106 
Rectangular pipe offset, 100 

related mathematics, 102 
Rectangular pyramids, frustums of, 

135 
Rectilinear figures, 1 
Register boxes, 143 

related mathematics, 145 
Related mathematics, 

ash barrel, 117 

ash pan, 155 

atomizing sprayer, 150 

boat pump, 127 

candy pan, 19 

chimney tube, 32 

conductor head, 178 

conical flower holder, 82 

conical roof flange, 91 

cup strainer, 164 

curved elbows, 106 

diagonal offset, 105 

dripping or roasting pans, 138 

elbows, 58, 97 

half-pint cup, 36 



Related mathematics, 

galvanized match box, 16 

garbage can, 42 

iron bread pan, 23 

liquid nieasures, 171 

oval to round transition, 193 

painter's pail, 40 

pitch top cover, 85 

rectangular flaring pan, 142 

rectangular pipe offset, 102 

register boxes, 145 

rotary ash sifter, 160 

sheet metal cleat, 13 

short handled dipper, 167 

split header, 213 

sprinkling can, 123 

tangent tees and tee joints, 76 
Return and face miters, 173 
Revolution, solids of, 80, 97 
Right cone, 80 
Roasting pan, 136 

related mathematics, 138 
Roof flange, 129 
Roof flange, conical, 89 

related mathematics, 91 
Rotary ash sifter, 156 

related mathematics, 160 
Round elbow, oval to, 216 
Round transition, oval to, 191 

related mathematics, 193 
Round transition, square to, 188 

Scalene cone, 186 

frustum of, 204 

triangulation of, 185 
Scale, triangular, 6 
Schedules of materials, 150 
Scoop, 46 

Sections, developed and extended, 215 
Sections, developments by, 203 
Sections formed by cutting planes, 

119 
Sector, area of, 82 
Sheet metal cleat, 12 

related mathematics, 13 
Sheet metal drafting, 1 
Short handled dipper, 165 

related mathematics, 167 



INDEX 



233 



Sifter, ash, rotary, 156 

related mathematics, 160 
Solids, combination of various, 147 
Solids of revolution, 80, 97 
Split header, square to round, 210 

related mathematics, 213 
Sprayer, atomizing, 148 

related mathematics, 150 
Sprinkling can, 120 

related mathematics, 123 
Square pipe, transition with second 

piece of elbow, 196 
Square return miter, 174 
Square to round split header, 210 

related mathematics, 213 
Square, T-, 3 

Square to round transition, 188 
Standard cuts of pipe, 61 
Standard sizes for liquid measures, 

171 
Strainer, cup, 162 

related mathematics, 164 

Tables, 

deduction for small end cuts, 60 

standard big end cuts for pipe 
and elbows, 61 

standard sizes for flaring liquid 
measures, 171 

United States standard gage for 
sheet and plate iron and 
steel, 1893, 228 

weights per square foot of gal- 
vanized and black sheets, 62 



Tangent tee at right angles, 69 

not at right angles, 75 
Tangent tees and tee joints, 76 

related mathematics, 76 
Tee joint at right angles, 66 

not at right angles, 72 
Three-piece rectangular elbow, 96 

related mathematics, 97 
Titles, 9 

Transition, oval pipe and second piece 
of elbow, 200 

square pipe and second piece of 
• elbow, 196 

square to round, 188 

triangulation of pieces, 195 
Transition, oval to round, 191 

related mathematics, 193 
Trapezoid, area of, 23 
Triangles, 4 
Triangular scale, 6 
Triangulation of scalene cones, 185 

of transition pieces, 195 
T-square, 3 
Two-piece elbow, 49 
Two-piece 60° elbow, 52 

Vegetable parer, 86 
Vertical lines, 5 
Volume of cylinder, 40 
Volume of frustum, 167 

Window cap, 182 
Wired cylinders, 29 
Witness dimensions, 42 



INDEX OF PROBLEMS 



NO. OP . 

PROBLEM PAGE 

1. Laying out a Metal Cleat 12 

2. Galvanized Match Box 14 

3. Candy Pan . 17 

4. Bread Pan .21 

5. Galvanized Chimney Tube 30 

6. Half-pint Cup 34 

7. Painter's Pail 38 

8. Garbage Can 41 

9. The Scoop 46 

10. Two-piece Elbow 49 

11. Two-piece 60° Elbow 52 

12. Four-piece 90° Elbow 55 

13. Long Radius Elbows 57 

14. Backset Method 63 

15. Tee Joint at Right Angles 66 

16. Tangent Tee at Right Angles 69 

17. Tee Joint Not at Right Angles 72 

18. Tangent Tee Not at Right Angles . 75 

19. Conical Flower Holder 80 

20. Pitch Top Cover 83 

21. Vegetable Parer 86 

22. Conical Roof Flange 89 

23. Apron for Conical Roof Flange 93 

24. Three-piece Rectangular Elbow 96 

25. Rectangular Pipe Offset 100 

26. Diagonal Offset 103 

27. Curved Elbow in Rectangular Pipe . ."-^^P*^ , 106 

28. Ash Barrel ... 110 

29. Sprinkhng Can 120 

30. Boat Pump 125 

31. Roof Flange 129 

32. Dripping or Roasting Pan 136 

33. Rectangular Flaring Pans 140 

34. Register Boxes 143 

35. Atomizing Sprayer 148 

36. Ash Pan with Semicircular Back 153 

37. Rotary Ash Sifter . . . . • 156 

235 



236 INDEX OF PROBLEMS 

NO. OF 
PROBLEM PAGE 

38. Cup Strainer 162 

39. Short Handled Dipper 165 

40. Liquid Measures 169 

41. Square Return Miter 174 

42. Conductor Head ... 177 

43. Face Miters 180 

44. Window Cap 182 

45. Scalene Corie 186 

46. Square to Round Transition . 188 

47. Oval to Round Transition 191 

48. Transition between a Square Pipe and the Second Piece of an Elbow 19G 

49. Transition between an Oval Pipe and the Second Piece of an Elbow . 200 

50. The Frustum of a Scalene Cone 204 

51. Center Offset Boot 207 

52. Square to Round Split Header 210 

53. Oval to Round Elbow 216 

54. Breeches 221 

55. Coal Hod , . . 225 



